1 Introduction
Every Borel subset of an uncountable Polish metric space X is a result of the following inductive procedure, whose steps are enumerated by countable ordinals: for
$\alpha =1$
, we take all open subsets of X and their complements (closed subsets of X). We continue by transfinite induction: if
$\beta <\omega _1$
is a countable ordinal and we completed our procedure for all ordinals
$0<\alpha <\beta $
, then at the step
$\beta $
, we add all possible countable unions (or intersections) of sets obtained so far and their complements. This completes level
$\beta $
of the procedure. So at the step
$\alpha =2$
, we add all
$G_{\delta }$
and
$F_\sigma $
subsets, which are neither open nor closed, and in the next step, we obtain all sets that are countable intersections of
$F_\sigma $
sets and all countable unions of
$G_\delta $
sets, which are neither
$G_\delta $
nor
$F_\sigma $
themselves. This introduces a natural hierarchy of Borel subsets of X that reflects their descriptive complexity. It is known that for an uncountable Polish space, these levels do not collapse: at each level, there appear new sets (sets that do not belong at any lower level of the hierarchy). Determining the level where a ‘naturally arising’ or ‘non-ad hoc’ Borel set appears in the hierarchy becomes a challenging problem. Only a few concrete examples of Borel sets are known to be located above the third level.
Here, we study the Borel complexity of sets appearing naturally in ergodic theory, dynamics, number theory and fractal geometry. We develop tools that allow us to determine the exact position of many Borel sets examined in the literature. We show that without compactness, some sets studied in topological dynamics become completely analytic (completely analytic sets are, in some sense, the simplest instances of non-Borel sets).
We state our main result in the framework of a dynamical system given by iteration of a continuous map T on a Polish metric space X. In this setting, the sets of interest consist of points whose orbits have prescribed statistical behaviour with respect to a bounded continuous observable
$\varphi \colon X\to \mathbb {R}$
. Here, by ‘statistical behaviour of an orbit of
$x\in X$
’, we mean the asymptotic behaviour of the sequence of Birkoff averages of
$\varphi $
along the orbit of x. In the classical topological dynamics, one usually assumes that the space X is compact, but some expansions, most notably, continued fraction expansions, come from dynamical systems acting on non-compact Polish spaces, so we work in this greater generality. We refer the reader to [Reference Grivaux and Matheron15–Reference Iommi and Velozo17, Reference Sarig31] for related problems for dynamical systems (linear operators, countable Markov chains) on non-compact phase spaces. We study the following questions.
Problem. Given a continuous map T from a Polish space X to itself and a T-invariant measure
$\mu $
, what is the complexity of the set
$G_\mu $
of generic points for the system (
$G_\mu $
is defined precisely below)? More generally, given a non-empty, closed and connected subset V of the space
${\mathcal {M}_{\mathit T}}(X)$
of all T-invariant Borel probability measures on X endowed with the weak
$^*$
topology, what is the Borel complexity of the set
$G(V)$
of all points x in X whose orbits generate V, that is, the set of limit points of a sequence
$\mathcal {E}(x,N)$
equals V, where
$\mathcal {E}(x,N)$
denotes the normalized counting measure concentrated on
$x,T(x),\ldots ,T^{N-1}(x)$
. Finally, given a bounded continuous function
$\varphi \colon X\to \mathbb {R}$
, we study Borel complexity of sets of those
$x\in X$
where the sequence of Birkhoff averages
$$ \begin{align*} A_k\varphi(x)=\frac1k\sum_{j=0}^{k-1}\varphi(T^j(x))\end{align*} $$
converges, as
$k\to \infty $
, to a given value
$\alpha \in \mathbb {R}$
. We also examine the set of points where this sequence does not converge for some or all bounded continuous function(s) (the irregular set).
Our initial interest in these questions comes from a result of H. Ki and T. Linton, who answered a question posed by Kechris and showed in [Reference Ki and Linton20] that for any integer
$r \geq 2$
, the set of numbers that are normal in base r is a
$\boldsymbol {\Pi }^0_3$
-complete set (a
$\boldsymbol {\Pi }^0_3$
set is a countable intersection of
$F_\sigma $
sets, the general definitions are given in §2). Recall that we say that a real number
$\xi $
is normal in base r if in its r-adic expansion, every block of digits of length k appears with asymptotic frequency
$1/r^k$
. After [Reference Ki and Linton20], many authors have studied the Borel complexity of various sets related to normal numbers, and have extended Ki and Linton’s result in different directions. Becher, Heiber and Slaman [Reference Becher, Heiber and Slaman4] settled a conjecture of A. S. Kechris by showing that the set of absolutely normal numbers (a real number is absolutely normal if it is normal to all bases
$r\ge 2$
) is
$\boldsymbol {\Pi }^0_3$
-complete. Furthermore, Becher and Slaman [Reference Becher and Slaman5] proved that the set of numbers normal in at least one base is
$\boldsymbol {\Sigma }^0_4$
-complete. See also [Reference Airey, Jackson and Mance2, Reference Beros6]. Also, sets of interest in dynamics are sometimes complete at levels past the Borel hierarchy. For example, in [Reference Jackson, Mance and Roth18], a dynamical system in the unit square is constructed for which the set of special
$\alpha $
-limit points is
$\boldsymbol {\Pi }^1_1$
-complete.
It is well known that a real number
$\xi $
being normal in base r is equivalent to the sequence
$(r^k \xi )_k $
being uniformly distributed mod
$1$
(cf. [Reference Kuipers and Niederreiter21]). The latter statement can be rephrased as
$\xi \bmod 1$
is a generic point for the Lebesgue measure on the unit circle
$\mathbb {T}=\mathbb {R}/\mathbb {Z}$
under the action of the map
$\mathbb {T}\to \mathbb {T}$
given by
$x\mapsto r x\bmod 1$
. Thanks to this dynamical point of view, in [Reference Airey, Jackson, Kwietniak and Mance1], we were able to study the Borel complexity of sets of normal numbers in several numeration systems. We employed a unified treatment for r-ary expansions, continued fraction expansions,
$\beta $
-expansions and generalized GLS-expansions. In fact, we considered a dynamical system given by subshifts generated by all possible expansions of numbers in
$[0,1]$
in full-shifts over an at most countable set of digits (alphabet). In that setting, normal numbers with respect to a given expansion method correspond to generic points of an appropriately selected shift-invariant measure on a subshift containing all possible expansions. It turns out that for these subshifts, the set of generic points for any shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a
$\boldsymbol {\Pi }^0_3$
-complete set). The crucial dynamical feature we used was a feeble form of specification. All expansions named above generate subshifts with this specification property.
Our aim is to extend the results of [Reference Airey, Jackson, Kwietniak and Mance1]. We generalize [Reference Airey, Jackson, Kwietniak and Mance1] in two directions.
First, we consider arbitrary (in particular, non-symbolic) dynamical systems on Polish spaces. In [Reference Airey, Jackson, Kwietniak and Mance1], only symbolic systems were considered. The list of systems that are covered by the current paper contains all systems with the specification property. The specification property was introduced by Bowen in his paper on Axiom A diffeomorphisms [Reference Bowen8]. The most prominent examples in that class are topologically mixing uniformly hyperbolic systems, in particular, hyperbolic toral automorphisms. Other important classes of non-symbolic dynamical systems that also have the specification property are mixing interval maps or, more generally, graph maps. Of course, we can also apply our results to any product of the systems with the specification property (symbolic or non-symbolic). We refer the reader to [Reference Kwietniak, Ła̧cka and Oprocha22] for a discussion of the specification property and its many variants as well as their significance in dynamics. However, for the main results of our paper, only a certain weak form of the specification property (coined the strong approximate product structure, abbreviated as SAPS) is needed. It implies the feeble specification property used in [Reference Airey, Jackson, Kwietniak and Mance1] for symbolic systems. Examples of non-symbolic topological dynamical systems without the specification property, but with the strong approximate product structure, are provided by automorphism of a compact abelian group that is ergodic with respect to the Haar measure, but does not have the specification property (see [Reference Kwietniak, Ła̧cka and Oprocha22] for more details on these examples).
Second, we determine the descriptive complexity of sets of points defined by placing restrictions on statistical distribution of their orbits with respect to all bounded continuous observables. By statistical distribution of the orbits, we understand the asymptotic behaviour of a sequence of Birkhoff averages along an orbit. In [Reference Airey, Jackson, Kwietniak and Mance1], only sets of generic points were considered. Our level of generality still allows us to study dynamical systems leading to various expansions considered in [Reference Airey, Jackson, Kwietniak and Mance1], but it also contains many more examples: we are now able to study the descriptive complexity of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their expansions. To apply our results in other branches of mathematics, it remains to explain how to reduce the set we want to examine to the relevant dynamical form. So far, most studies concentrated on the Hausdorff dimension of these sets, see, for example, Barreira, Saussol and Schmeling [Reference Barreira, Saussol and Schmeling3], Eggleston [Reference Eggleston12], Fan, Feng and Wu [Reference Fan, Feng and Wu13], Olsen [Reference Olsen27], Usachev [Reference Usachev33] or Łuczak [Reference Łuczak25], Wang and Wu [Reference Wang and Wu34] for continued fraction expansions, but these references are only the top of an iceberg. We also note that the Borel complexity of some sets of similar origin was recently studied by Olsen [Reference Olsen28] and Tomaszewski [Reference Tomaszewski32].
Simple examples of systems with ergodic invariant measures whose generic points form Borel sets of low complexity explain why here, as well as in [Reference Airey, Jackson, Kwietniak and Mance1], we need additional assumptions to achieve the maximal complexity. If X is the unit circle and T is an irrational rotation, then every point is generic for the Lebesgue measure
$\unicode{x3bb} $
, so
$G_\unicode{x3bb} $
is clopen. However, if
$X=[0,1]$
and
$T(x)=x^2$
for
$0\le x \le 1$
, then T has two ergodic invariant measures, namely Dirac-point measures
$\delta _0$
and
$\delta _1$
concentrated at fixed points
$0$
and
$1$
. We clearly have
$G_{\delta _0}=[0,1)$
is open but not closed, and
$G_{\delta _1}=\{1\}$
is closed but not open. It turns out that the right assumption to impose on the system is a form of the specification property mentioned above.
Our main result (Theorem 7.4) determines the Borel complexity of the set of generic points
$G_\mu $
for Polish dynamical systems with the SAPS. Assuming that the set of invariant measures is non-trivial (that is, there are at least two invariant measures), the set of generic points
$G_\mu $
is
$\boldsymbol {\Pi }^0_3$
-complete. In Theorem 7.6, we also determine the upper bounds for Borel complexity of similarly defined sets of dynamical significance (see §4). As a corollary, we get that under our fairly general assumptions, the sets of interest always have the maximal possible complexity, but we sometimes have to assume compactness of the phase space to achieve that (see Corollaries 7.5 and 7.7). The necessity of the compactness assumption is demonstrated by an example of a (non-compact) dynamical system for which
$G(V)$
(here, V may consist of a single measure
$\mu $
) is
$\boldsymbol {\Pi }^1_1$
-complete (the example is constructed to prove Theorem 8.1). Thus, we see a dichotomy in the complexity of
$G(V)$
depending on whether X is compact. We also construct a minimal compact symbolic system (subshift) with an invariant ergodic measure
$\mu $
such that the set
$G_\mu $
is
$\boldsymbol {\Pi }_3^0$
-complete. This example shows that the specification property is sufficient to guarantee maximal complexity of the sets of generic points, but by no means necessary.
1.1 Organization of the paper
After reviewing the notation and terminology in §2, we discuss the specification property and its weakening in §3. In §4, we define various sets described by the statistical behaviour of Birkhoff averages, and in §5, we prove that all these sets are (co-)analytic, some are always Borel and all are always Borel when the space X is compact. In other words, in §5, we gather upper bounds for the descriptive complexity of our sets of interest. These computations hold in general, that is, they do not require specification hypotheses on the system. Section 6 contains some technical auxiliary results needed for the subsequent proofs. In §7, we state and prove our main results (Theorems 7.4 and 7.6) regarding lower bounds assuming our variant of the specification property (the strong approximate product structure). In §8, we present a (non-compact) dynamical system such that
$G(V)$
is
$\boldsymbol {\Pi }^1_1$
-complete, where
$V=\{\mu \}$
for some invariant measure
$\mu $
compact. Finally, in §9, we give an example of a minimal compact subshift having an invariant ergodic measure
$\mu $
whose corresponding set of generic points
$G_\mu $
is
$\boldsymbol {\Pi }_3^0$
-complete.
2 Vocabulary/definitions/notation
2.1 General
Throughout this paper,
$\omega =\{0,1,2,\ldots \}$
,
$\mathbb {N}$
stands for the set of positive integers, the variables
$i,j,k,\ell ,m,n$
are integers and X always denotes a Polish topological space. Note that we are not assuming compactness of X in general; any compactness assumptions on X will be stated explicitly as needed.
By a Polish dynamical system, which we also just call a dynamical system, we mean a pair
$(X,T)$
where
$T\colon X\to X$
is a continuous map. We stress that whenever we consider a Polish dynamical system
$(X,T)$
, we tacitly assume that X is equipped with a fixed compatible complete metric
$\rho $
. If X is non-compact, then various properties we will be considering, such as the specification property and its variants, depend on
$\rho $
in a sense that changing the metric to an equivalent one may affect these properties. For an example where the specification property is lost with a change of metric, see [Reference Kwietniak and Ubik23, Proposition 7.1] (in other words, for non-compact spaces, the specification property is no longer an invariant for topological conjugacy). For compact spaces, the choice of metric is irrelevant.
2.2 Borel, analytic and co-analytic sets
We recall some basic notions from descriptive set theory that we use to measure the complexity of sets in Polish spaces. The collection of Borel sets
$\mathcal {B}(X)$
, for X a topological space, is the smallest
$\sigma $
-algebra containing the open sets.
We let
$\boldsymbol {\Sigma }^0_1$
be the collection of open sets and 
be the collection of closed sets. In general, for
$\alpha <\omega _1$
, we let
$\boldsymbol {\Sigma }^0_\alpha $
be the collection of countable unions
$A=\bigcup _n A_n$
, where each
$A_n \in \boldsymbol {\Pi }^0_{\alpha _n}$
for some
$\alpha _n<\alpha $
. We also let 
. Equivalently,
$A\in \boldsymbol {\Pi }^0_\alpha $
if
$A=\bigcap _n A_n$
, where
$A_n\in \boldsymbol {\Sigma }^0_{\alpha _n}$
and each
$\alpha _n<\alpha $
. We also set
$\boldsymbol {\Delta }^0_\alpha = \boldsymbol {\Sigma }^0_\alpha \cap \boldsymbol {\Pi }^0_\alpha $
. In particular,
$\boldsymbol {\Delta }^0_1$
is the collection of clopen sets. In classical terminology,
$\boldsymbol {\Sigma }^0_2$
is the collection of
$F_\sigma $
sets and
$\boldsymbol {\Pi }^0_2$
is the collection of
$G_\delta $
sets. For any topological space,
$\mathcal {B}(X)=\bigcup _{\alpha <\omega _1} \boldsymbol {\Sigma }^0_\alpha = \bigcup _{\alpha <\omega _1}\boldsymbol {\Pi }^0_\alpha $
. A basic fact (see [Reference Kechris19]) is that for any uncountable Polish space X, all of the classes
$\boldsymbol {\Delta }^0_\alpha $
,
$\boldsymbol {\Sigma }^0_\alpha $
,
$\boldsymbol {\Pi }^0_\alpha $
, for
$\alpha <\omega _1$
, are distinct. We say a set
$A\subseteq X$
is
$\boldsymbol {\Sigma }^0_\alpha $
(respectively
$\boldsymbol {\Pi }^0_\alpha $
) hard if
$A \notin \boldsymbol {\Pi }^0_\alpha $
(respectively
$A\notin \boldsymbol {\Sigma }^0_\alpha $
). This says that A is ‘no simpler’ than a
$\boldsymbol {\Sigma }^0_\alpha $
set. We say that A is
$\boldsymbol {\Sigma }^0_\alpha $
-complete if
$A\in \boldsymbol {\Sigma }^0_\alpha \setminus \boldsymbol {\Pi }^0_\alpha $
, that is,
$A \in \boldsymbol {\Sigma }^0_\alpha $
and A is
$\boldsymbol {\Sigma }^0_\alpha $
hard. This says that A is exactly at the complexity level
$\boldsymbol {\Sigma }^0_\alpha $
. Likewise, A is
$\boldsymbol {\Pi }^0_\alpha $
-complete if
$A\in \boldsymbol {\Pi }^0_\alpha \setminus \boldsymbol {\Sigma }^0_\alpha $
.
The Borel sets lie at the base of another hierarchy, the projective hierarchy. The
$\boldsymbol {\Sigma }^1_1$
(or analytic) sets are the images of Borel sets (via continuous functions from a Polish space to a Polish space). The same collection results if one uses Borel images instead of continuous images and one can also take just images of closed sets. The
$\boldsymbol {\Pi }^1_1$
(or co-analytic) sets are the complements of the analytic sets.
We define
$\boldsymbol {\Delta }^1_1=\boldsymbol {\Sigma }^1_1\cap \boldsymbol {\Pi }^1_1$
. The pointclasses
$\boldsymbol {\Sigma }^1_1$
and
$\boldsymbol {\Pi }^1_1$
are closed under countable unions and intersections, and so contain all the Borel sets. A fundamental result of Suslin (see [Reference Kechris19]) says that in any Polish space,
$\mathcal {B}(X)=\boldsymbol {\Delta }^1_1$
. One can continue to define the
$\boldsymbol {\Sigma }^1_n$
and
$\boldsymbol {\Pi }^1_n$
sets for
$n \geq 2$
, forming the projective hierarchy, but we will not consider any classes beyond
$\boldsymbol {\Pi }^1_1$
and
$\boldsymbol {\Sigma }^1_1$
in this paper.
Also, all of the collections
$\boldsymbol {\Delta }^0_\alpha $
,
$\boldsymbol {\Sigma }^0_\alpha $
,
$\boldsymbol {\Pi }^0_\alpha $
,
$\boldsymbol {\Pi }^1_1$
and
$\boldsymbol {\Sigma }^1_1$
are pointclasses, that is, they are closed under inverse images of continuous functions.
The levels of the Borel (and projective) hierarchy can be used to calibrate the descriptive complexity of a set. Determining an upper bound on the complexity of the set A generally involves writing a condition defining A that shows that it is in the appropriate place in the hierarchy. Establishing lower bounds is generally more difficult. One technique is the method of ‘Wadge reduction’, whereby a set
$C\subseteq Y$
(Y a Polish space), which is known to be hard for a certain level
$\boldsymbol {\Gamma }$
of the hierarchy, is ‘reduced’ to the set A. By this, we mean we have a continuous function
$f\colon Y\to X$
such that
$C=f^{-1}(A)$
. Since
$\boldsymbol {\Gamma }$
is a pointclass, this shows that A is
$\boldsymbol {\Gamma }$
-hard.
2.3 Invariant measures, generic points
We denote the set of all Borel probability measures on X by
$\mathcal {M}(X)$
and equip it with the weak
$^*$
topology, which, in our setting, is known to be Polish [Reference Kechris19, §17.E]. Recall that
$\mu _n$
converges to
$\mu $
in the weak
$^*$
topology if and only if for every
$f \in C_b(X)$
(that is, f is a bounded, continuous, real-valued function on X), we have
$\int f\, d\mu _n \to \int f\, d\mu $
. If X is compact, then
$\mathcal {M}(X)$
is also compact [Reference Kechris19, Theorem 17.22]. Although our main results are of a topological nature, it is convenient for us to fix, once and for all, a particular metric D for
$\mathcal {M}(X)$
. We choose the Prohorov metric associated to our fixed metric
$\rho $
on X and given by
$$ \begin{align} D(\mu,\nu)= \inf\{\varepsilon>0: \mu(A)\le \nu(A^\varepsilon)+\varepsilon\text{ for all Borel sets }A\subset X\}, \end{align} $$
where
$\mu ,\nu \in \mathcal {M}(X)$
and
$A^\varepsilon $
denotes the
$\varepsilon $
-neighbourhood of A, that is,
$$ \begin{align*}A^\varepsilon=\{y\in X:\rho(x,y)<\varepsilon\text{ for some }x\in A\}.\end{align*} $$
Although equation (1) does not seem to yield a symmetric function, we do obtain a metric since we consider only Borel probability measures. The empirical measure
$\mathcal {E}(\bar x)$
of a sequence
$\bar x=\langle x_1,\ldots ,x_k\rangle \in X^k$
, where
$k>0$
is the normalized sum of the Dirac (point-mass), measures along points in the sequence, that is,
$$ \begin{align*} \mathcal{E}(\bar x)=\frac{1}{k}\sum_{p=1}^{k}\delta_{x_p}, \end{align*} $$
where
$\delta _x$
stands for the Dirac measure concentrated at
$x\in X$
. The following lemma gathers some elementary properties of the Prokhorov metric that we use throughout the paper (often without further reference). The proof is straightforward.
Lemma 2.1. Let
$k>0$
,
$\mu _1,\mu _2\in \mathcal {M}(X)$
and
$\alpha ,\beta \in [0,1]$
.
-
(i) For any
$x,y\in X$
, we have
$$ \begin{align*} D(\delta_x, \delta_y) = \min\{\rho(x,y),1\}. \end{align*} $$
-
(ii) If
$0\le \varepsilon <1$
and
$\bar {x}=\langle x_1,\ldots ,x_k\rangle ,\bar {y}=\langle y_1,\ldots ,y_k\rangle \in X^k$
satisfy then
$$ \begin{align*} \rho(x_1,y_1)\le\varepsilon, \ldots, \rho(x_k,y_k)\le\varepsilon, \end{align*} $$
$D(\mathcal {E}(x),\mathcal {E}(y))\le \varepsilon $
.
-
(iii) For
$j=1,2$
, we have
$D(\alpha \mu _1+(1-\alpha )\mu _2,\mu _j)\le D(\mu _1,\mu _2)$
. -
(iv) If
$\nu _1,\nu _2\in \mathcal {M}(X)$
and
$\delta =\max \{D(\mu _j,\nu _j)\colon j=1,2\}$
, then
$$ \begin{align*} D(\alpha\mu_1+(1-\alpha)\mu_2,\beta \nu_1+(1-\beta)\nu_2) \le |\alpha-\beta|+\delta. \end{align*} $$
-
(v) If
$n,m> 0$
,
$\bar {x}=\langle x_1,\ldots ,x_m\rangle \in X^m,\bar {y}=\langle y_1,\ldots ,y_n\rangle \in X^n$
,
$d_1=D(\mathcal {E}(\bar {x}),\mu _1)$
and
$d_2=D(\mathcal {E}(\bar {y}),\mu _2)$
, then setting
$\bar {z}=\langle z_1,\ldots ,z_{m+n}\rangle \in X^{m+n}$
, where we have
$$ \begin{align*}z_j=\begin{cases} x_j & \mbox{if } 1\le j \le m, \\ y_{j-m} & \mbox{if } m< j \le m+n, \end{cases}\end{align*} $$
$$ \begin{align*} D\Big(\mathcal{E}(\bar{z}),\frac{m}{m+n}\mu_1+\frac{n}{m+n}\mu_2\Big) \le \max\{d_1,d_2\}. \end{align*} $$
Given a Polish dynamical system
$(X,T)$
, we say that
$\mu \in \mathcal {M}(X)$
is T-invariant if
$\mu (T^{-1}(A))=\mu (A)$
for every Borel set
$A \subseteq X$
. We let
${\mathcal {M}_{\mathit T}}(X)$
be the set of T-invariant Borel probability measures on X. If X is compact, then
${\mathcal {M}_{\mathit T}}(X)$
must be non-empty, but this is no longer true if X is just Polish, even if T exhibits non-trivial recurrence (see [Reference Grivaux and Matheron15, p. 374]). A measure
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
is ergodic if for every Borel set
$A\subseteq X$
satisfying
$T^{-1}(A)\subseteq A$
, it holds that
$\mu (A)$
is either
$0$
or
$1$
. We write
${\mathcal {M}_{\mathit T}^e}(X)$
for the set of T-invariant ergodic measures. The set
${\mathcal {M}_{\mathit T}}(X)$
is always convex, so
${\mathcal {M}_{\mathit T}^e}(X)$
is always non-empty if
${\mathcal {M}_{\mathit T}}(X)\neq \emptyset $
.
Definition 2.2. We say that
$x \in X$
is a generic point for
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
if the sequence of empirical measures, whose nth entry is given by
$$ \begin{align} \mathcal{E}(x,n)= \frac{1}{n}(\delta_{T^0(x)}+\delta_{T^1(x)}+\cdots +\delta_{T^{n-1}(x)}),\end{align} $$
converge in the weak
$^*$
topology on
$\mathcal {M}(X)$
to
$\mu $
as
$n\to \infty $
. We let
$G_\mu =G_\mu (X,T)$
denote the set of generic points for
$\mu $
.
If
$\mu $
is ergodic, then it follows from the ergodic theorem that
$\mu $
-almost every
$x \in X$
is a generic point for
$\mu $
. We will not, however, be assuming
$\mu $
is necessarily ergodic in our results, and so the existence of generic points is not automatic.
The notion of generic point can be viewed as generalizing the notion of a normal number. If we take
$X=r^\omega $
, T the shift map on
$r^\omega $
and
$\mu $
the standard Bernoulli
$({1}/{r},\ldots ,{1}/{r})$
measure on
$r^\omega $
, then
$x \in r^\omega $
is a generic point if and only if x is the base r expansion of a normal number.
3 Specification property and its weakening
To present our modification of the specification property, we develop a terminology allowing us to compare various specification-like notions. It is inspired by [Reference Kwietniak, Ła̧cka and Oprocha22]. Given a Polish dynamical system
$(X,T)$
and
$x\in X$
, we define the orbit of x to be the set
$O(x)=\{T^n(x):n\in \omega \}$
. Although the orbit is a set, it is customary to consider it as a sequence
$\langle T^n(x):n\in \omega \rangle $
. Using this interpretation and given
$a,b\in \omega $
with
$a\le b$
, we define an orbit segment of x from a to b (synonymously, over
$[a,b]$
) to be the sequence
$T^{[a,b]}(x)=\langle T^a(x),\ldots ,T^b(x)\rangle $
. Every orbit segment (up to the choice of a and b) is determined by its initial point
$y=T^a(x)$
and length
$n=b-a+1>0$
, and hence, without loss of generality, an orbit segment may be identified with a pair
$\langle y,n\rangle $
in
$X\times \omega \setminus \{0\}$
. We will use these two formulations concurrently. Note that in the following definition and throughout the rest of this paper, we use a convention that a sum with empty range of indices (an empty sum), for example,
$\sum _{i=1}^{0}\ldots $
equals
$0$
.
Definition 3.1. A specification of rank
$k>0$
is a finite sequence
$\xi $
of k orbit segments, that is,
$\xi \in (X\times \omega \setminus \{0\})^k$
. Let
$\varepsilon>0$
. We say that a point
$y\in X$
$\varepsilon $
-close shadows (or traces or follows) a specification
$\xi =\langle \langle x_1,n_1\rangle ,\ldots ,\langle x_k,n_k\rangle \rangle \in (X\times \omega \setminus \{0\})^k$
if there are integers (called gaps)
$s_1,\ldots ,s_{k-1}\in \omega $
such that for every
$1\le j\le k$
, we have
$$ \begin{align*} \rho( T^{\sum_{i=1}^{j-1} (n_i+s_i)+t}(y),T^t(x_j))<\varepsilon\quad\text{for }0\le t<n_j. \end{align*} $$
The number
$L=n_k+\sum _{i=1}^{k-1} (n_i+s_i)$
is the number of iterates of y required to trace
$\xi $
. We may also say that the orbit segment
$\langle y,L\rangle $
traces
$\varepsilon $
-close the specification
$\xi $
.
Definition 3.2. We say that T has the specification property if for every
$\varepsilon>0$
, there is a constant
$N_\varepsilon \in \omega $
such that for each
$k>0$
and every specification
$\xi \in (X\times \omega \setminus \{0\})^k$
, there is a point y that traces it
$\varepsilon $
-close with all gaps of size
$\leq N_\varepsilon $
.
To state our weakening of the specification property, we will need the following notation. By
$\rho _\Lambda (x,y)$
, we denote the Bowen distance between
$x,y\in X$
along a finite set
$\Lambda \subseteq \omega $
given by
$$ \begin{align*} \rho_\Lambda(x,y)=\max_{j\in\Lambda} \rho(T^j(x),T^j(y)). \end{align*} $$
The Bowen ball of radius
$\varepsilon>0$
, along
$\Lambda $
and centred at
$x\in X$
, is defined as
$$ \begin{align*} B_\Lambda(x,\varepsilon)=\{y\in X:\rho_\Lambda(x,y)<\varepsilon\}. \end{align*} $$
When
$\Lambda \subseteq \{0,1,\ldots ,n-1\}$
for some
$n\in \mathbb {N}$
and
$|\Lambda |/n$
is close to
$1$
, then the Bowen ball
$B_\Lambda (x,\varepsilon )$
can be seen as a set of those points y in X whose orbits almost
$\varepsilon $
-trace the orbit segment
$\langle x,n\rangle $
. Here, ‘almost’ means that the number of ‘errors’, that is, those
$0\le j<n$
for which
$\rho ( T^j(x),T^j(y))\ge \varepsilon $
, is small. If
$\Lambda =\{0,1,\ldots ,n-1\}$
for some
$n\in \mathbb {N}$
, then no errors are allowed and we denote a Bowen ball
$B_\Lambda (x,\varepsilon )$
as
$B_n(x,\varepsilon )$
.
We will now introduce a weakening of the specification property for a dynamical system. Although a huge variety of generalizations of the specification property are available in the literature [Reference Kwietniak, Ła̧cka and Oprocha22], we find it convenient to introduce yet another one. The newly defined property suffices to prove our main results.
Definition 3.3. We say that a point
$y\in X$
$(\varepsilon ,\delta _1,\delta _2)$
-approximately traces a specification
$\xi =(\langle x_j,n_j\rangle )_{j=1}^k$
if:
-
(i)
$y\in B_{n_1}(x_1,\varepsilon )$
; -
(ii) for each
$2\le j \le k$
, there is a set
$\Lambda _{j}\subseteq \{0,\ldots ,n_{j}-1\}$
with
$|\Lambda _{j}|\ge (1 - \delta _1) n_{j} $
and an integer
$0\le s_{j-1}\le \delta _2 n_{j}$
such that
$$ \begin{align*} T^{(\sum_{i=1}^{j-1}s_i+n_i)}(y)\in B_{\Lambda_j}(x_j,\varepsilon). \end{align*} $$
Definition 3.4. A Polish dynamical system
$(X, T )$
has the strong approximate product structure if the following holds. Given
$\varepsilon , \delta _1, \delta _2> 0$
, there exists a non-negative integer
$N=N(\varepsilon ,\delta _1,\delta _2)$
such that for any specification
$\xi =(\langle x_j,n_j\rangle )_{j=1}^k$
with
$n_j \ge N$
for
$1 \le j \le k$
, there exists a point
$y \in X$
that
$(\varepsilon ,\delta _1,\delta _2)$
-approximately traces
$\xi $
.
Remark 3.5. We will also say that a point y
$\varepsilon $
-approximately traces a specification
$\xi $
if it
$(\varepsilon , \varepsilon , \varepsilon )$
-approximately traces
$\xi $
as in Definition 3.3. We define
$N(\varepsilon ) = N(\varepsilon , \varepsilon , \varepsilon )$
, where
$N(\cdot ,\cdot ,\cdot )$
is as in Definition 3.4.
Thus, strong approximate product structure implies that, for any
$\varepsilon , \delta _1,\delta _2>0$
and any specification
$\xi = \langle (x_1,n_1),\ldots , (x_n,n_k)\rangle $
with all
$n_i$
sufficiently large, we can find a point y that
$\varepsilon $
-traces the first orbit segment
$(x_1,n_1)$
and
$\varepsilon $
-traces the segments
$(x_i,n_i)$
for
$i\ge 2$
with at most
$\delta _1 n_i$
errors, allowing gaps of sizes at most
$\delta _2 n_i$
between the segments
$(x_{i-1}, n_{i-1})$
and
$(x_i,n_i)$
.
The strong approximate product structure is similar to and implies the approximate product structure of Pfister and Sullivan [Reference Pfister and Sullivan30]. The difference between these two notions is that in Definition 3.3,
$\varepsilon $
-approximate-tracing allows no errors in the first orbit segment. Below, we picture the implications between the SAPS and specification properties investigated by other authors: the specification property (SP), the weak specification property (WSP) and approximate product structure (APS), see [Reference Kwietniak, Ła̧cka and Oprocha22] for more details:
$$ \begin{align*} \textrm{SP} \Rightarrow \textrm{WSP} \Rightarrow \textrm{SAPS} \Rightarrow \textrm{APS}. \end{align*} $$
Although definition of the specification property looks prohibitively restrictive, there are many classes of systems exhibiting a variant of that property. Familiar examples of symbolic systems (subshifts) that have the specification property are mixing shifts of finite type and mixing sofic shifts. There are many other symbolic systems with the specification property. All topologically mixing graph maps (in particular, interval maps) have the specification property. The specification property is closely related with hyperbolicity: every transitive uniformly hyperbolic diffeomorphism of a compact connected manifold has the specification property. Specification, or its variants, also holds for systems of algebraic origin. Following Lind [Reference Lind24], we say that a toral automorphism is quasi-hyperbolic if its associated matrix A has no roots of unity as eigenvalues. Lind [Reference Lind24, Theorems (ii)–(iii)] and Marcus [Reference Marcus26] proved that all quasi-hyperbolic toral automorphisms must satisfy weak specification. As the strong approximate product structure implies the feeble specification property used in [Reference Airey, Jackson, Kwietniak and Mance1], our results also apply to the systems considered in [Reference Airey, Jackson, Kwietniak and Mance1], most notably to
$\beta $
-shifts.
4 Sets defined by conditions on statistical behaviour of their orbits
Generalizing the notion of a generic point, we define
$V(x)$
as the set of all limit points in
$\mathcal {M}(X)$
of the sequence
$(\mathcal {E}(x,n))_{n\ge 0}$
. Thus, if
$x\in G_\mu $
, then
$V(x)=\{\mu \}$
, and the converse is true if X is compact. It is well known that
$V(x)$
is a closed connected subset of
${\mathcal {M}_{\mathit T}}(X)$
, which is necessarily non-empty provided that X is compact.
Given a set
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
, we can relate it to
$V(x)$
and define sets of points whose orbits exhibit particular statistical behaviour.
Definition 4.1. If
$V\subseteq \mathcal {M}(X)$
, we let
$G^V=G^V(X,T)=\{ x \in X \colon V\subseteq V(x) \}$
. Similarly, let
$G_V=G_V(X,T)=\{ x \in X \colon V(x) \subseteq V\}$
. We set
$G(V)= G^V \cap G_V$
. Finally, we let 
.
We refer to the set
$G(V)$
as the set of generic points for V. We caution the reader that in the non-compact case, it is not necessarily true that
$G_\mu =G(\{ \mu \})$
(see Remark 8.2).
Definition 4.2. Given
$\varphi \in \mathcal {C}_b(X)$
,
$k>0$
and
$y\in X$
, we define the kth ergodic average of
$\varphi $
along the orbit of y by
$$ \begin{align} A_k\varphi(y)=\frac{1}{k}\sum_{j=0}^{k-1} \varphi(T^j(y))=\int \varphi\, d \mathcal{E}(y,k). \end{align} $$
Definition 4.3. Given
$\varphi \in \mathcal {C}_b(X)$
, we say that a point
$x\in X$
is Birkhoff regular for
$\varphi $
if the sequence of ergodic averages
$A_k\varphi (x)$
converges as
$k\to \infty $
. For
$\alpha \in \mathbb {R}$
, we define the
$\alpha $
-level sets for Birkhoff averages (
$\alpha $
-Birkoff regular points) of
$\varphi $
by
$$ \begin{align*} B_{\varphi}(\alpha)=\Big\{x\in X: \lim_{k\to\infty} A_k\varphi(x)=\alpha\Big\}. \end{align*} $$
The set of all Birkhoff regular points for
$\varphi $
is denoted
$$ \begin{align*} B_\varphi(X,T)=\bigcup_{\alpha\in\mathbb{R}}B_\varphi(\alpha). \end{align*} $$
We also define
${\mu }(\varphi ):=\int _X\varphi \ \textit {d}\mu $
.
Definition 4.4. The irregular set for
$\varphi $
is the set
$I_\varphi (X,T)$
consisting of all points
$y\in X$
such that the sequence of ergodic averages
$(A_k\varphi (y))_{k\in \mathbb {N}}$
does not converge. The irregular set
$I(X, T)$
of the Polish dynamical system
$(X,T)$
is the union of
$I_\varphi (X,T)$
when
$\varphi $
runs over all bounded continuous real-valued functions on X.
Thus, for each
$\varphi \in \mathcal {C}_b(X)$
, we have a partition
$$ \begin{align*} X=I_\varphi(X,T)\cup \bigcup_{\alpha\in\mathbb{R}}B_\varphi(\alpha). \end{align*} $$
By the Birkhoff ergodic theorem, the irregular set is universally null:
$\mu (I_\varphi (X,T))=\mu (I(X, T))=0$
for every T-invariant Borel probability measure
$\mu $
on X and
$\varphi \in C_b(X)$
. The complement of the irregular set, denoted
$Q(X,T)$
, is called the quasi-regular set of
$(X,T)$
. If X is compact, then it is easy to see that every quasi-regular point must be also generic for some T-invariant measure (cf. [Reference Oxtoby29]). It seems to be a little known fact that each quasi-regular point must be generic for some T-invariant measure also in the general situation, that is, for every Polish metric space. For example, Oxtoby in [Reference Oxtoby29, §7] (following Fomin) adds an additional condition to the definition of quasi-regular point for a Borel automorphism to get a similar result.
Fact 4.5. We have
$Q(X,T)=\bigcup _{\mu \in {\mathcal {M}_{\mathit T}}(X)} G_\mu $
.
Proof. Note that if x is generic for
$\mu $
, then the definition of weak
$^*$
convergence and equation (3) imply that
$A_k\varphi (y)\to \mu (\varphi )=\int \varphi \, \textit {d}\mu $
as
$k\to \infty $
, so every generic point is
$\mu (\varphi )$
-Birkhoff regular for every
$\varphi \in \mathcal {C}_b(X)$
. In particular, each generic point is quasi-regular. However, the space of Borel measures on X is weakly sequentially complete by [Reference Bogachev7, Theorem 8.7.1]. Here, weak sequential completeness means that every fundamental sequence converges (cf. [Reference Bogachev7, §8.1]). A sequence
$(\mu _n)_{n\ge 1}$
of Borel probability measures is fundamental if for every bounded continuous function f, the sequence
$\int f \textit {d}\mu _n$
is Cauchy. Therefore, if x is quasi-regular, then the sequence
$(\mathcal {E}(x,n))_{n\ge 1}$
is fundamental and so converges to a Borel probability measure
$\mu $
on X, which is easily seen to be T-invariant.
Note that for all Polish dynamical systems
$(X,T)$
, for each
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
and for each
$\varphi \in \mathcal {C}_b(X)$
, we have
$$ \begin{align*} G_\mu\subseteq Q(X,T) \subseteq B_\varphi(X,T) \quad\text{and} \quad I_\varphi(X,T)\subseteq I(X,T). \end{align*} $$
5 Upper bounds
We will study which sets, defined by placing conditions on the statistical behaviour of orbits, are Borel (assuming that X is compact if necessary). Recall that
$(X,T)$
is a Polish dynamical system and let
$(f_n)$
be a sequence of bounded continuous real-valued functions on X that generate the weak
$^*$
topology on
$\mathcal {M}(X)$
(cf. [Reference Kechris19, Theorem 17.19]). That is,
$\mu _k\hspace{-1pt} \to\hspace{-1pt} \mu $
as
$k\to \infty $
if and only if for each n, we have
$\int f_n\, d\mu _k \to \int f_n\, d\mu $
as
$k\to \infty $
. Recall also that
$A_k\varphi $
stands for the kth ergodic average of
$\varphi $
along the orbit of y, see equation (3).
Fact 5.1. If
$\varphi \in \mathcal {C}_b(X)$
and
$\alpha \in \mathbb {R}$
, then the set
$B_\varphi (\alpha )=\{x\in X:\lim _{k\to \infty }A_k\varphi (x)=\alpha \}$
of
$\alpha $
-Birkhoff regular points is a
$\boldsymbol {\Pi }^0_3$
set.
Proof. It is easy to see that
$$ \begin{align*} B_\varphi(\alpha)=\bigcap_{m=1}^\infty\bigcup_{K=1}^\infty\bigcap_{k=K}^\infty \{y\in X\colon | A_k\varphi(y)-\alpha|\le 1/m\}.\\[-45pt] \end{align*} $$
Note that the proof of Fact 5.1 actually yields that for every sequence of real-valued continuous functions
$(f_k)_{k=1}^\infty $
on X and
$\alpha \in \mathbb {R}$
, the set of all points
$x\in X$
such that the sequence
$(f_k(x))_{k=1}^\infty $
converges to
$\alpha $
is a
$\boldsymbol {\Pi }^0_3$
set.
Fact 5.2. If
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
, then the set
$G_\mu $
of
$\mu $
-generic points is a
$\boldsymbol {\Pi }^0_3$
set.
Proof. A point
$y\in X$
belongs to
$G_\mu $
if and only if for every n, the sequence
$A_k f_n(y)$
converges to
$\mu (f_n)=\int f_n\,\text {d}\mu $
as
$k\to \infty $
. Thus,
$$ \begin{align*} G_\mu=\bigcap_{n=0}^\infty B_{f_n}(\mu(f_n)). \end{align*} $$
By Fact 5.1,
$G_\mu $
is a
$\boldsymbol {\Pi }^0_3$
set.
Fact 5.3. If
$\varphi \in \mathcal {C}_b(X)$
, then the irregular set
$I_\varphi (X,T)$
is a
$\boldsymbol {\Sigma }^0_3$
set and
$B_\varphi (X,T)= X\setminus I_\varphi (X,T)$
is a
$\boldsymbol {\Pi }^0_3$
set.
Proof. A point
$x\in X$
is irregular for
$\varphi $
if and only if
$$ \begin{align*} \liminf_{k\to\infty}A_k\varphi(x)<\limsup_{k\to\infty}A_k\varphi(x). \end{align*} $$
Therefore,
$$ \begin{align*} I_\varphi(X,T)=\bigcup_{N=1}^\infty \bigcap_{K=1}^\infty \bigcup_{k= K}^\infty \bigcup_{\ell= K}^\infty \{x\in X\colon A_k\varphi(x)+1/N<A_\ell \varphi(x)\}. \end{align*} $$
By continuity, the set
$\{x\in X\colon A_k\varphi (x)+1/N<A_\ell \varphi (x)\}$
is open in X for any choice of
$\varphi $
and
$k,\ell ,m$
. It follows that
$I_\varphi (X,T)$
is a
$\boldsymbol {\Sigma }^0_3$
set.
Fact 5.4. The irregular set
$I(X,T)$
is a
$\boldsymbol {\Sigma }^0_3$
set and the quasi-regular set
$Q(X,T)= X\setminus I(X,T)$
is a
$\boldsymbol {\Pi }^0_3$
set.
Proof. Note that by Fact 4.5, we have
$x\in Q(X,T)$
if and only if x is generic for some
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
. The latter is equivalent to the sequence
$(\mathcal {E}(x,n))_{n\ge 1}$
being a Cauchy sequence in the Prohorov metric D (starting with the complete metric
$\rho $
on X, we obtain a complete metric D on
$\mathcal {M}(X)$
, see [Reference Garling14, Corollary 18.6.3]). Hence,
$$ \begin{align*} Q(X,T)=\bigcap_{m=1}^\infty\bigcup_{N=1}^\infty \bigcap_{k= N}^\infty \bigcap_{\ell= N}^\infty \{ x \in X \colon D(\mathcal{E}(x, k), \mathcal{E}(x,\ell)) \le 1/m \}. \end{align*} $$
Note that for any
$n \geq 1$
, the map
$x \mapsto \mathcal {E}(x, n)$
is continuous. Thus, the set
$\{ x \in X \colon D(\mathcal {E}(x, k), \mathcal {E}(x,\ell )) \le 1/m \}$
is closed for any choice of the parameters
$k,\ell $
and m. Therefore,
$Q(X,T)$
is a
$\boldsymbol {\Pi }^0_3$
set.
Fact 5.5. If
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is closed, then
$G^V=\{x\in X\colon V\subseteq V(x)\}$
is a
$\boldsymbol {\Pi }^0_2$
set.
Proof. Let
$V_0$
be a countable dense subset of V. Then,
$$ \begin{align*} G^V = \bigcap_{\nu \in V_0} G^{\{ \nu \} } = \bigcap_{\nu \in V_0} \bigcap_{N = 1}^\infty \bigcap_{m = 1}^\infty \bigcup_{n = N}^\infty \{ x \in X \colon D(\mathcal{E}(x, n), \nu) < 1/m \}. \end{align*} $$
Note that for any
$n \geq 1$
, the map
$x \mapsto \mathcal {E}(x, n)$
is continuous. Thus, the set
$\{ x \in X \colon D(\mathcal {E}(x, n), \nu ) < 1/m \}$
is open for any choice of the parameters n and
$\nu $
. Thus,
$G^V$
is a
$\boldsymbol {\Pi }^0_2$
set.
Recall that a point
$x \in X$
is said to have maximal oscillation if
$V(x) = {\mathcal {M}_{\mathit T}}(X)$
[Reference Denker, Grillenberger and Sigmund10, Definition 21.17]. Taking
$V = {\mathcal {M}_{\mathit T}}(X)$
in Fact 5.5, we immediately obtain the following corollary.
Fact 5.6. The set of points with maximal oscillation is a
$\boldsymbol {\Pi }^0_2$
set.
Fact 5.7. If
$U\subseteq {\mathcal {M}_{\mathit T}}(X)$
is open, then the set 
is analytic (is
$\boldsymbol {\Sigma }^1_1$
).
Proof. Let
$R=\{(x,\mu )\in X\times \mathcal {M}(X):\mu \in V(x)\}\subseteq X\times {\mathcal {M}_{\mathit T}}(X)$
. We have
$$ \begin{align*} R=\bigcap_{m=1}^\infty \bigcap_{N=1}^\infty \bigcup_{n=N}^\infty \{(x,\mu) \in X\times {\mathcal{M}_{\mathit T}}(X) :D(\mu,\mathcal{E}(x,n))<1/m\}. \end{align*} $$
Clearly, the set
$\{(x,\mu ) \in X\times {\mathcal {M}_{\mathit T}}(X) :D(\mu ,\mathcal {E}(x,n))<1/m\}$
is open in
$X\times {\mathcal {M}_{\mathit T}}(X)$
for every
$n,m>0$
, so
$\{(x,\mu )\in X\times \mathcal {M}(X):\mu \in V(x)\}\subseteq X\times {\mathcal {M}_{\mathit T}}(X)$
is Borel. Since 
is a projection of the set
$R\cap (X\times U)$
onto the first coordinate, we get that 
is
$\boldsymbol {\Sigma }^1_1$
(see [Reference Kechris19, p. 86]).
We have the following as an immediate corollary.
Fact 5.8. For every closed
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
, the set
$G_V$
is co-analytic (is
$\boldsymbol {\Pi }^1_1$
).
Proof. Recall that 
and apply Fact 5.7.
Fact 5.9. Assume X is compact. If
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is closed, then
$G_V\hspace{-1pt}=\hspace{-1pt}\{x\in X\hspace{-1pt}\colon\hspace{-1pt} V(x)\hspace{-1pt}\subseteq\hspace{-1pt} V\}$
is a
$\boldsymbol {\Pi }^0_3$
set. If
$U \subseteq {\mathcal {M}_{\mathit T}}(X)$
is open, then 
is a
$\boldsymbol {\Sigma }^0_3$
set.
Proof. Observe that
$$ \begin{align*} V(x) \subseteq V \Leftrightarrow & \; D(\mathcal{E}(x, n), V) \to 0 \quad\textrm{as } n \to \infty \\ \Leftrightarrow & \; x \in \bigcap_{m = 1}^\infty \bigcup_{N = 1}^\infty \bigcap_{n = N}^\infty \{ x \in X : D(\mathcal{E}(x, n), V) \le 1/m \}. \end{align*} $$
Here, we employed the standard notation
$D(\mathcal {E}(x, n), V) = \inf _{v \in V} D(\mathcal {E}(x, n), v)$
. We use the compactness of X in the first equivalence; without compactness, only the implication
$\Leftarrow $
holds. Since the map
$x \mapsto \mathcal {E}(x, n)$
is continuous for every
$n \geq 1$
, the sets
$\{ x \in X : D(\mathcal {E}(x, n), V) \le 1/m \}$
are closed. Thus,
$G_V$
is a
$\boldsymbol {\Pi }^0_3$
set. For the second part of the fact, note that 
for every open set
$U\subseteq X$
.
Fact 5.10. If
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is closed, then the set
$G(V)$
is co-analytic (is
$\boldsymbol {\Pi }^1_1$
). If, in addition, X is compact, then
$G(V)$
is a
$\boldsymbol {\Pi }^0_3$
set.
The following question is related to those considered in this paper.
Question. Given a dynamical system
$(X,T)$
,
$\varphi \in \mathcal {C}_b(X)$
and
$A\subseteq \mathbb {R}$
, what is the exact complexity of the set
$$ \begin{align*} B(\varphi,A)=\bigcup_{\alpha\in A}B_\varphi(\alpha)? \end{align*} $$
6 Auxiliary results
The next three results record some basic estimates that we will use in the proof of our main result. To formulate these results, we have to extend our terminology.
Recall that the empirical measure
$\mathcal {E}(\bar x)$
of a sequence
$\bar x=\langle x_1,\ldots ,x_k\rangle \in X^k$
, where
$k>0$
, is the normalized sum of the Dirac (point-mass) measures along points in the sequence, that is,
$$ \begin{align*} \mathcal{E}(\bar x)=\frac{1}{k}\sum_{p=1}^{k}\delta_{x_p}. \end{align*} $$
Given a specification
$\xi =(\langle x_j,n_j\rangle )_{j=1}^k\in (X\times \omega \setminus \{0\})^k$
, we define the empirical measure of
$\xi $
denoted by
$\mathcal {E}(\xi )$
as the empirical measure of the sequence
$\operatorname {\mathrm {seq}}(\xi )\in X^{\sum _{j=1}^kn_j}$
obtained by concatenation of the orbits segments in
$\xi $
, that is,
$\mathcal {E}(\xi )=\mathcal {E}(\operatorname {\mathrm {seq}}(\xi ))$
, where
$$ \begin{align*} \operatorname{\mathrm{seq}}(\xi)=\langle x_1,T(x_1),\ldots, T^{n_1-1}(x_1),\ldots,x_k,T(x_k),\ldots,T^{n_k-1}(x_k)\rangle\in X^{\sum_{j=1}^kn_j}. \end{align*} $$
Note that if
$\xi =\langle (x,n)\rangle $
is a single orbit segment for some
$n>0$
, then we have
$$ \begin{align*} \mathcal{E}(\xi)=\mathcal{E}(\langle x,T(x),\ldots, T^{n-1}(x)\rangle)= \frac{1}{n}\sum_{j=0}^{n-1}\delta_{T^j(x)}=\mathcal{E}(x,n), \end{align*} $$
where
$\mathcal {E}(x,n)$
is the empirical measure defined by equation (2).
Lemma 6.1. Let
$\gamma ',\delta ',\varepsilon '\ge 0$
. Assume that
$\bar x=\langle x_1,\ldots ,x_k\rangle \in X^k$
and
$\bar y=\langle y_1,\ldots ,y_\ell \rangle \in X^\ell $
, where
$0<k\le \ell \le (1+\delta ')k$
. If there exists j with
$(1-\gamma ')k\le j\le k$
and subsequences
$\bar x'=\langle x_{a_1},\ldots ,x_{a_j}\rangle \in X^j$
and
$\bar y'=\langle y_{b_1},\ldots ,y_{b_j}\rangle \in X^j$
with
$1\le a_1 <\cdots < a_j\le k$
and
$1\le b_1 <\cdots < b_j\le \ell $
such that
$\rho (x_{a_i},y_{b_i})\le \varepsilon '$
for
$1\le i\le j$
, then
$$ \begin{align*} D(\mathcal{E}(\bar x),\mathcal{E}(\bar y)) \le 2\gamma'+\delta'+\varepsilon'. \end{align*} $$
Proof. By the triangle inequality,
$$ \begin{align} D(\mathcal{E}(\bar{x}),\mathcal{E}(\bar{y})) \leq D((\mathcal{E}(\bar{x}), \mathcal{E}(\bar x')) +D(\mathcal{E}(\bar x'), \mathcal{E}(\bar y'))+D(\mathcal{E}(\bar y'), \mathcal{E}(\bar{y})). \end{align} $$
For each
$1\leq i \leq j$
, we have from the definition of D that
$D(\delta _{x_{a_i}},\delta _{y_{b_i}})\le \varepsilon '$
. It follows that
$D(\mathcal {E}(\bar x'), \mathcal {E}(\bar y'))\le \varepsilon '$
.
Next, observe that for every Borel set
$A\subseteq X$
, we have
$$ \begin{align*} \mathcal{E}(\bar{x})(A) \le \frac{j}{k}\mathcal{E}(\bar{x}')(A) + \frac{k-j}{k} \le \mathcal{E}(\bar{x}')(A^{\gamma'}) + \gamma', \end{align*} $$
which means that
$D(\mathcal {E}(\bar {x}), \mathcal {E}(\bar {x}'))\leq \gamma '$
. By our assumptions,
$(1 - \gamma ' - \delta ') \ell \le j \le \ell $
; this implies
$D(\mathcal {E}(\bar {y}), \mathcal {E}(\bar y')) \leq \gamma ' + \delta '$
by a similar reasoning. Plugging all the mentioned inequalities into equation (4) finishes the proof.
We have the following immediate corollary.
Corollary 6.2. Let
$k>0$
and
$\delta _1,\delta _2,\varepsilon \ge 0$
. If a point y
$(\varepsilon ,\delta _1,\delta _2)$
-approximately traces a specification
$\xi =(\langle x_j,n_j\rangle )_{j=1}^k\in (X\times \omega \setminus \{0\})^k$
with gaps
$s_1,\ldots ,s_{k-1}\in \omega $
satisfying
$s_j\le \delta _2 n_j$
for
$1\le j<k$
and
$$ \begin{align*} L=n_k+\sum_{i=1}^{k-1}(s_i+n_i), \end{align*} $$
then
$D(\mathcal {E}(\xi ),\mathcal {E}(y,L))\le 2\delta _1+\delta _2+\varepsilon $
.
Proof. We apply Lemma 6.1 with
$\gamma '=\delta _1$
and
$\delta '=\delta _2$
, and
$\varepsilon '=\varepsilon $
to
$\bar {x}=\operatorname {\mathrm {seq}}(\xi )$
and
$\bar {y}=\langle y,L\rangle $
, where y is a point that
$(\varepsilon ,\delta _1,\delta _2)$
-approximately traces
$\xi $
(see Definition 3.3).
Lemma 6.3. If
$(X,T)$
is a Polish dynamical system with the strong approximate product structure and
$\mu $
is a T-invariant measure on X, then for each
$\varepsilon>0$
, there exists a positive integer q such that for every
$n>0$
, there is
$x\in X$
satisfying
$D(\mu ,\mathcal {E}(x,nq))<\varepsilon $
.
Proof. Assume that
$\mu $
is a T-invariant (possibly non-ergodic) probability measure. It is well known that
$\mu $
can be approximated by affine combinations of finite collections of ergodic measures with rational coefficients, that is, for every
$\varepsilon>0$
, there is a measure
$\nu \in {\mathcal {M}_{\mathit T}}(X)$
such that
$$ \begin{align} D(\mu,\nu)<\varepsilon/8\quad \text{and}\quad\nu=\sum_{i=1}^{k} q_i \nu_i, \end{align} $$
where
$k>0$
and
$\nu _1,\ldots ,\nu _k\in {\mathcal {M}_{\mathit T}}(X)$
are ergodic,
$q_1,\ldots ,q_k\in \mathbb {Q}$
are positive, and
$q_1+\cdots +q_k=1$
. Let Q be a positive integer such that for some
$p_1,\ldots ,p_k$
, we have
$q_i=p_i/Q$
for
$i=1,\ldots ,k$
. We choose
$z_1,\ldots ,z_k$
such that
$z_i$
is generic for
$\nu _i$
. Next, let
$K\geq N(\varepsilon /16)$
(here,
$N(\cdot )$
is as in Remark 3.5) be large enough so that
$$ \begin{align} D(\mathcal{E}(z_i,K),\nu_i)<\varepsilon/8\quad \text{for }i=1,\ldots,k. \end{align} $$
For each
$n>0$
, we define a specification
$\xi $
by
$$ \begin{align*} \xi=(\langle x^{\prime}_j,K\rangle)_{j=1}^{nQ}\in (X\times \omega\setminus\{0\})^{nQ}, \end{align*} $$
where each
$z_i$
appears
$np_i$
times on the list
$x^{\prime }_1,\ldots ,x^{\prime }_{nQ}$
, that is,
$$ \begin{align*} \langle x^{\prime}_j\rangle_{j=1}^{nQ}=\langle \underbrace{z_1,\ldots,z_1}_{np_1 \text{ times}}, \underbrace{z_2,\ldots,z_2}_{np_2 \text{ times}},\,\ldots\,,\underbrace{z_k,\ldots,z_k}_{np_k \text{ times}}\rangle. \end{align*} $$
By definition of
$\xi $
, we have
$$ \begin{align} \mathcal{E}(\xi)=\sum_{i=1}^k q_i\mathcal{E}(z_i,K). \end{align} $$
Furthermore, equations (5) and (6), together with convexity of the distance on the space of measures, yield
$$ \begin{align*} D(\mu,\mathcal{E}(\xi))<\varepsilon/4. \end{align*} $$
Let x be a point that
$\varepsilon /16$
-approximately traces the specification
$\xi $
(one more time, recall Remark 3.5) and let L be the number of iterates required to do so. Then, Corollary 6.2 yields
$D(\mathcal {E}(\xi ),\mathcal {E}(x,L))<\varepsilon /4$
and, hence,
$D(\mu ,\mathcal {E}(x,L))<\varepsilon /2$
because of equation (7).
Since
$nKQ\le L\le (1+\varepsilon /16)nKQ$
, we use Lemma 6.1 with
$\gamma '=0$
,
$\delta '=\varepsilon /16$
and
$\varepsilon '=0$
to get that setting
$q=QK$
, we have
$D(\mathcal {E}(x,nq),\mathcal {E}(x,L))\le \varepsilon /16$
, which ends the proof.
Remark 6.4. Lemma 6.3 shows, assuming
$(X,T)$
has the strong approximate product structure, that the set of measures of the form
$\mathcal {E}(x,n)$
for
$x\in X$
and
$n>0$
is weak
${}^*$
dense in
${\mathcal {M}_{\mathit T}}(X)$
in the sense that every
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
is the weak
${}^*$
limit of a sequence
$\mu _k=\mathcal {E}(x_k,n_k)$
(in general, the measures
$\mathcal {E}(x,n)$
are not T-invariant).
7 Main results
In this section, we prove our main results about the complexity of the set of generic points for systems satisfying the strong approximate product structure (Theorems 7.4 and 7.6). If the system
$(X, T)$
is compact, the set
${\mathcal {M}_{\mathit T}}(X)$
is non-empty (a theorem of Bogolyubov and Krylov). If X is compact and
${\mathcal {M}_{\mathit T}}(X)$
consists of a single element
$ \mu $
, then
$X = G_\mu $
. Thus, for compact X, the statistical behaviour of the system is trivial if there exist less than two T-invariant measures. For this reason, in this section, we mostly work under the assumption that
$|{\mathcal {M}_{\mathit T}}(X) | \ge 2$
. It should be kept in mind, however, that for general Polish X, the set
${\mathcal {M}_{\mathit T}}(X)$
may be empty.
Example 7.1. Let
$\hat X$
be a compact metric space and
$\hat {T}\colon \hat {X}\to \hat {X}$
be a continuous map with the specification property. For example, we can take the compact unit interval,
$\hat {X}=[0,1]$
and
$\hat {T}$
, to be the tent map
$\hat {T}(x)=1-|1-2x|$
. Let X be the set of points with the maximal oscillation. It was noted by Sigmund that X is a dense
$G_\delta $
subset of
$\hat {X}$
, and X is a T-invariant set. Hence, restricting
$\hat {T}$
to X, we obtain a Polish dynamical system that does not have any invariant measures, since
$I(X,T)=X$
. It is easy to see that
$(X,T)$
has the specification property. It shows that we need to assume that
$|{\mathcal {M}_{\mathit T}}(X) | \ge 2$
.
In Lemma 7.2, we show (assuming that
$(X,T)$
has the strong approximate product structure) that given a sequence of T-invariant measures
$(\mu _n)_{n \ge 1}$
, satisfying
$D(\mu _n, \mu _{n+1}) \to 0$
, one can find a point
$y \in X$
and piecewise constant (constant on finite sets of consecutive j terms) non-decreasing function
$j\mapsto n(j)$
such that the sequence of empirical measures
$\mathcal {E}(y, j)$
is asymptotic to the sequence
$\mu _{n(j)}$
(the distance
$D(\mathcal {E}(y,j),\mu _{n(j)})$
converges to
$0$
as
$j\to \infty $
).
Part (i) of Theorem 7.4 gives the exact complexity of the set of generic points
$G_\mu $
for a Polish dynamical system
$(X,T)$
with the strong approximate product structure satisfying
$|{\mathcal {M}_{\mathit T}}(X) | \ge 2$
. More generally, part (ii) of Theorem 7.4, uses the same proof method to obtain lower bounds for the complexity of the set
$G(V)$
, where
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is a non-trivial, closed, connected subset of
${\mathcal {M}_{\mathit T}}(X)$
. If X is compact, then
$G_\mu =G(\{ \mu \})$
and so, in this case, part (i) follows directly from the statement of part (ii) of Theorem 7.4. In this section, we also prove lower bounds on the complexity of the sets
$G^V, G_V$
and other sets introduced in §5.
Lemma 7.2. Let
$(X, T)$
be a Polish dynamical system with the strong approximate product structure.
-
(i) Given a sequence
$(\mu _n)_{n \geq 1}$
in
${\mathcal {M}_{\mathit T}}(X)$
such that
$D(\mu _n, \mu _{n+1}) \to 0$
, there exists a point
$y \in X$
and a sequence
$L_n \nearrow \infty $
such that (8)where
$$ \begin{align} D(\mathcal{E}(y, j), \mu_{n(j)}) \to 0 \quad \textrm{as } j \to \infty, \end{align} $$
$n(j)$
denotes the unique integer satisfying
$L_{n(j) - 1} \le j < L_{n(j)}$
. In particular, this implies
$V(y) = \{ \textrm {limit points of the sequence } (\mu _k)_{k \geq 1} \}$
.
-
(ii) Given any
$y_0 \in X$
and
$\varepsilon> 0$
, we can define y in part (i) so that
$y \in B(y_0, \varepsilon )$
. -
(iii) If
$y^{1}$
and
$y^{2}$
are points obtained as in part (i) for sequences of measures
$( \mu ^{1}_n ) _{n \ge 1}$
and
$( \mu ^{2}_n ) _{n \ge 1}$
such that for some
$N\ge 1$
, we have that
$\mu ^{1}_i = \mu ^{2}_i$
for
$1\le i \le N$
, then
$\rho (y^{1}, y^{2}) \le 1 / 2^N$
.
Proof. Fix
$0< \varepsilon < 1$
. We will define sequences of integers
$(K_n)_{n\ge 0}$
and points
$(x_n)_{n\ge 0} \subseteq X$
that will be parameters for our construction. For each
$n\ge 0$
, we use Lemma 6.3 to pick
$x_n \in X$
and
$K_n$
so that the following hold:
$$ \begin{align} K_n\ge N(\varepsilon/2^{n+1}), \end{align} $$
$$ \begin{align} D(\mathcal{E}(x_n,K_n),\mu_n) \le \varepsilon/2^{n+1}. \end{align} $$
Here,
$N(\varepsilon /2^{n+1})$
is defined as in Remark 3.5.
Next, we will define
$(\ell _n)_{n\ge 1}, (L_n)_{n \ge 0}$
and
$(y_n)_{n \ge 0} \subseteq X$
. We begin with
$L_0 = K_1$
and
$y_0$
an arbitrary point in X. For all
$n \ge 0$
, we will have
$$ \begin{align} L_{n-1} \ge n K_n. \end{align} $$
We proceed inductively, for
$n \ge 1$
:
-
• pick
$\ell _n$
such that
$\ell _n K_n \ge (n + 1) K_{n+1}$
and
$\ell _n K_n \ge n L_{n-1}$
; -
• define the specification
$\xi _n$
to consist of the pair
$(y_{n-1}, L_{n-1})$
followed by the pair
$(x_n, K_n)$
repeated
$\ell _n$
times; -
• define
$y_n$
to be a point that
$\varepsilon /2^{n+1}$
-approximately traces the specification
$\xi _n$
(see Definition 3.3 and Remark 3.5); -
• define
$L_n$
to be the length of orbit segment required for
$y_n$
to trace
$\xi _n$
. Note that condition (11) is satisfied, as
$L_n \ge \ell _n K_n \ge n K_{n+1}$
.
It follows from this definition that
$$ \begin{align} \rho(T^i(y_{n-1}), T^i(y_n)) \le \varepsilon / 2^{n+1} \quad \textrm{for } 0 \le i < L_{n-1}. \end{align} $$
In particular, the sequence
$(y_n)$
must converge to a limit y and we have
$$ \begin{align} \rho(T^i(y_{n-1}), T^i(y)) \le \varepsilon / 2^n \quad \textrm{for } 0 \le i < L_{n-1}. \end{align} $$
We will now examine the empirical measures
$\mathcal {E}(y, j)$
for
$j>0$
. First, consider
$\mathcal {E}(y, L_n)$
for some integer n. By equation (13), we have
$$ \begin{align} D(\mathcal{E}(y, L_n), \mathcal{E}(y_n, L_n)) \le \varepsilon / 2^n. \end{align} $$
Recall that we have chosen
$y_n$
so that it
$\varepsilon /2^{n+1}$
-approximately traces
$\xi _n$
along the orbit segment of length
$L_n$
. Together with Lemma 6.1, this gives
$D(\mathcal {E}(y_n, L_n), \mathcal {E}(\xi _n)) < 4 \varepsilon /2^{n+1} = \varepsilon / 2^{n-1}$
.
Define
$\bar {\xi _n}$
to be the specification consisting of
$\ell _n$
repetitions of pair
$(x_n, K_n)$
. Thus,
$\bar {\xi _n}$
only differs from
$\xi _n$
by the lack of initial
$(y_{n-1}, L_{n-1})$
segment. We defined
$\ell _n$
so that
$\ell _n K_n \ge n L_{n-1}$
and, thus, an application of Lemma 6.1 gives
$$ \begin{align} D(\mathcal{E}(\xi_n), \mathcal{E}(\bar{\xi_n})) \le 1/n. \end{align} $$
Finally, notice that
$$ \begin{align} D(\mathcal{E}(\bar{\xi_n}), \mu_n) = D(\mathcal{E}(x_n, K_n), \mu_n) \le \varepsilon / 2^{n+1}. \end{align} $$
Combining equations (14), (15) and (16) gives
$$ \begin{align} D(\mathcal{E}(y, L_n), \mu_n) \le 1/n + 7 \varepsilon/2^{n+1} \to 0 \quad \textrm{as } n \to \infty. \end{align} $$
We are now prepared to estimate
$\mathcal {E}(y, j)$
for arbitrary
$j>0$
. There exists a unique integer
$n = n(j)$
such that
$L_{n-1} \le j < L_n$
. Recall the definition of
$y_n$
as a point that
$\varepsilon / 2^{n+1}$
-approximately traces
$\xi _n$
. Let
$s_1, \dots s_{\ell _n}$
be the lengths of gaps in this tracing, as in Definition 3.3. The orbit segment
$(y_n, L_n)$
is made up of orbit segments corresponding to the elements of
$\xi _n$
(interspersed by gaps of length
$s_i$
). Thus, the initial subsegment
$(y, j)$
fully contains some number
$b+1$
of those segments. Formally, b is the unique integer
$0 \le b < \ell _n$
such that
$$ \begin{align*} B := L_{n-1} + bK_n + \sum_{i < b} s_i \le j < L_{n-1} + (b+1)K_n + \sum_{i \le b} s_i. \end{align*} $$
We have
$j - B \le K_n + s_b \le (1 + \varepsilon / 2^{n+1}) K_n$
. We also have
$$ \begin{align} \frac{j - B}{B} \le \frac{(1 + \varepsilon / 2^{n+1}) K_n}{B} \le \frac{(1 + \varepsilon / 2^{n+1}) K_n}{L_{n-1}} \le \frac{1 + \varepsilon / 2^{n+1}}{n}, \end{align} $$
where the last inequality follows from equation (11). Next, note that
$y_n$
, over the orbit segment of length B,
$\varepsilon / 2^{n+1}$
-approximately traces the specification
$$ \begin{align*} \zeta = \langle (y_{n-1}, L_{n-1}), \underbrace{(x_n, K_n), \dots (x_n, K_n)}_{b \textrm{ times}} \rangle. \end{align*} $$
By the triangle inequality, we obtain
$$ \begin{align} D(\mathcal{E}(y, j), \mu_{n}) &\le D(\mathcal{E}(y, j), \mathcal{E}(y, B)) + D(\mathcal{E}(y, B), \mathcal{E}(y_n, B)) \notag \\ &\quad + D(\mathcal{E}(y_n, B), \mathcal{E}(\zeta)) + D(\mathcal{E}(\zeta), \mu_{n}). \end{align} $$
Let us examine the terms on the right-hand side of
$\le $
in equation (19):
-
•
$D(\mathcal {E}(y, j), \mathcal {E}(y, B)) \le (1 + \varepsilon / 2^{n+1}) / n$
by equation (18); -
•
$D(\mathcal {E}(y, B), \mathcal {E}(y_n, B)) \le \varepsilon / 2^{n+1}$
by equation (13); -
•
$D(\mathcal {E}(y_n, B), \mathcal {E}(\zeta )) \le 4\varepsilon / 2^{n+1}$
by
$\varepsilon / 2^{n+1}$
-approximately tracing and Lemma 6.1; -
•
$D(\mathcal {E}(\zeta ), \mu _{n}) \le \max \{ D(\mathcal {E}(y_{n-1}, L_{n-1}), \mu _{n}), D(\mathcal {E}(x_n, K_n), \mu _{n}) \}$
. Now:-
–
$D(\mathcal {E}(y_{n-1}, L_{n-1}), \mu _{n-1}) \to 0$
as
$n \to \infty $
, as shown by equation (17) and by our assumption
$D(\mu _n, \mu _{n-1}) \to 0$
; -
–
$D(\mathcal {E}(x_n, K_n), \mu _{n}) \le \varepsilon / 2^{n+1} \to 0$
as
$n \to \infty $
.
-
All of this, applied to the right-hand side of
$\le $
in equation (19), shows that
$$ \begin{align} D(\mathcal{E}(y, j), \mu_{n(j)}) \to 0 \quad \textrm{as } j \to \infty. \end{align} $$
It follows easily that
$V(y) \subseteq \{ \textrm {limit points of the sequence } (\mu _k)_{k \geq 1} \}$
. Indeed, if
$\mathcal {E}(y, j_k) \to \bar {\mu }$
for some subsequence
$(j_k)$
, then also
$\mu _{n(j_k)} \to \bar {\mu }$
. Conversely, if
$\mu _{n_k} \to \bar {\mu }$
for some
$\bar {\mu }$
, then
$\mathcal {E}(y, L_{n_k - 1}) \to \bar {\mu }$
.
It remains to address the ‘moreover’ part of the lemma statement.
-
(i) Regardless of the sequence
$(\mu _n)$
, by equation (13), we have
$\rho (y, y_0) \le \varepsilon / 2$
. -
(ii) Note that construction of
$y_1 \cdots y_n$
depends only on
$\mu _1 \cdots \mu _n$
. Consider another sequence
$(\mu _n')$
such that
$\mu _j = \mu _j'$
for
$j \le n$
. Let
$y'$
be the point obtained from this sequence. Then, since
$y_n = y_n'$
, we deduce from equation (13) that
$$ \begin{align*} \rho(y, y') \le \rho(y, y_n) + \rho(y_n', y') \le \varepsilon /2^{n+1} + \varepsilon /2^{n+1} = \varepsilon / 2^n.\\[-34pt] \end{align*} $$
We make the following observation.
Corollary 7.3. If
$(X,T)$
is a Polish dynamical system with the strong approximate product structure and
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is a non-empty closed connected set, then the set
$G(V)$
is dense in X. For any T-invariant measure
$\mu $
, the set
$G_\mu $
is dense.
Proof. For the first part, note that there exists a sequence
$(\mu _n)_{n \ge 1}$
such that
$D(\mu _n, \mu _{n+1}) \to 0$
and
$\{ \textrm {limit points of } (\mu _n)_{n \ge 1} \} = V$
. Thus, an application of Lemma 7.2 shows that
$G(V)$
is dense. Similarly, application of Lemma 7.2 to the constant sequence
$\mu _n = \mu $
shows that
$G_\mu $
is dense.
Theorem 7.4. If a Polish dynamical system
$(X,T)$
has the strong approximate product structure, then we have the following.
-
(i) If
$|{\mathcal {M}_{\mathit T}}(X)|\geq 2$
, then for any
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
, the set
$G_\mu $
of
$\mu $
generic points is
$\boldsymbol {\Pi }^0_3$
-complete.
Moreover, we have the following.
-
(ii) For every non-empty closed connected set
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
with non-empty complement
$U={\mathcal {M}_{\mathit T}}(X)\setminus V$
, the sets
$G_V$
and
$G(V)$
are
$\boldsymbol {\Pi }_3^0$
-hard (and so the set is
$\boldsymbol {\Sigma }_3^0$
-hard). Also, the set
$G^V$
is
$\boldsymbol {\Pi }^0_2$
-complete. The sets
$G_V$
,
$G^V$
,
$G(V)$
and are all dense in X.
is 
are all dense in X.Proof. We give the proof of statement (ii), and remark at the end how the proof shows statement (i). Let
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
be as in statement (ii) and let
$U={\mathcal {M}_{\mathit T}}(X) \setminus V$
. Let
$\mathcal {C}_3\subseteq \omega ^ \omega $
be the set
$\mathcal {C}_3=\{ \beta \in \omega ^ \omega \colon \beta (n) \to \infty \}$
. It is well known that
$\mathcal {C}_3$
is a
$\boldsymbol {\Pi }^0_3$
-complete set, see [Reference Kechris19]. We construct a continuous reduction map
$\pi \colon \omega ^\omega \to X$
such that if
$\beta \in \mathcal {C}_3$
, then
$\pi (\beta )\in G(V)\subseteq G_V$
, and if
$\beta \notin \mathcal {C}_3$
, then 
. This will show that
$G_V$
and
$G(V)$
are
$\boldsymbol {\Pi }^0_3$
-complete.
Take
$\nu \in U$
and let
$\mu '\in V$
. Consider the set
$E=\{0\le t\le 1: t\mu '+(1-t)\nu \in V\}$
. It is easy to see that
$E\neq \emptyset $
is closed,
$1\in E$
and
$0\notin E$
. Let
$t_0=\min E$
and
$\bar \mu =t_0\mu '+(1-t_0)\nu \in V$
. Hence,
$\alpha \bar \mu +(1-\alpha )\nu \in U$
for
$\alpha <1$
. Since V is closed and connected and
${\mathcal {M}_{\mathit T}}(X)$
is separable, there is a sequence
$(\mu _n)_{n\ge 1}$
of T-invariant measures satisfying
$$ \begin{align} D(\mu_n,\mu_{n+1})\to 0\quad\text{as } n\to\infty, \end{align} $$
$$ \begin{align} \bigcap_{N=1}^\infty\overline{\{\mu_n:n\ge N\}}=V. \end{align} $$
That is, the set of limit points of sequence
$(\mu _n)_{n \ge 1}$
is equal to V.
Select a sequence
$(n_k)_{k\ge 1}$
such that
$\mu _{n_k}\to \bar \mu $
as
$k\to \infty $
, and additionally,
$n_{k+1} - n_k \to \infty $
. Given a
$\beta \in \omega ^ \omega $
, we define a function
$$ \begin{align} \psi^\beta (j) = \frac{j - n_k}{n_{k+1} - n_k} \frac{1}{\beta(k) + 1} + \frac{n_{k+1} - j}{n_{k+1} - n_k} \frac{1}{\beta(k+1) + 1} \quad \textrm{for } n_k \le j < n_{k+1} \end{align} $$
and a sequence of measures
$$ \begin{align*} \mu^\beta_j = \psi^\beta (j) \nu + (1 - \psi^\beta (j) ) \mu_j. \end{align*} $$
Note that
$\psi ^\beta (n_k) = 1/\beta (k)$
for all k, and
$\psi ^\beta $
is ‘linear’ between
$n_k$
and
$n_{k+1}$
. Thus, we have
$$ \begin{align*} | \psi^\beta(j) - \psi^\beta(j+1) | \le \frac{1}{n_{k+1} - n_k} \to 0 \quad \textrm{as } j \to \infty. \end{align*} $$
Using Lemma 2.1(iii), we get that
$D(\mu ^\beta _j, \mu ^\beta _{j+1}) \to 0$
as
$j \to \infty $
. Thus, we can apply Lemma 7.2 to the sequence
$(\mu ^\beta _j)$
and obtain a point
$\pi (\beta )$
such that
$V(\pi (\beta )) = \{ \textrm {limit points of } \mu ^\beta _j \}$
. Let us consider cases.
-
• If
$\beta (n) \in \mathcal {C}_3$
, then
$\psi ^\beta (j) \to 0$
as
$j \to \infty $
. It follows that
$D(\mu ^\beta _j, \mu _j) \to 0$
and, consequently,
$V(\pi (\beta )) = \{ \textrm {limit points of } \mu ^\beta _j \} = \{ \textrm {limit points of } \mu _j \} = V$
. -
• If
$\beta (n) \not \in \mathcal {C}_3$
, then there exists a subsequence
$(m_k)$
such that
$\beta (m_k) = c$
. Then, and it follows that
$$ \begin{align*} \mu^\beta_{n_{m_k}} = \frac{1}{c} \nu + \frac{c-1}{c} \mu_{n_{m_k}} \to \frac{1}{c} \nu + \frac{c-1}{c} \bar{\mu} \not \in V \end{align*} $$
$V(\pi (\beta )) \not \subseteq V$
.
Finally, notice that
$\mu ^\beta _1, \dots \mu ^\beta _{n_k}$
depend only on
$\beta (1), \dots \beta (n_k)$
. Thus, the map
$\pi $
is continuous by Lemma 7.2(iii). This shows that
$G(V)$
and
$G_V$
are
$\boldsymbol {\Pi }^0_3$
-hard, and
$^U G$
is
$\boldsymbol {\Sigma }_3^0$
-hard. Note that Lemma 7.2(ii) shows that all those sets are dense.
Suppose now that
$| {\mathcal {M}_{\mathit T}}(X)|\geq 2$
and let
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
. We reapply the above proof for the case
$V=\{ \mu \}$
. Since
$|{\mathcal {M}_{\mathit T}}(X)|\geq 2$
,
$U={\mathcal {M}_{\mathit T}}(X)\setminus V \neq \emptyset $
. We can take the measures
$(\mu _n)$
of the previous proof to be all equal to
$\mu $
. We also let
$\nu \in U={\mathcal {M}_{\mathit T}}(X)\setminus V$
. For
$\beta \in \mathcal {C}_3$
, the previous argument shows in this case that not only is
$\pi (\beta ) \in G(\{ \mu \})$
, but in fact,
$\pi (\beta ) \in G_\mu $
. This follows from equation (8) in Lemma 7.2(i). For
$\beta \notin \mathcal {C}_3$
, the previous argument gives that
$\pi (\beta ) \notin G(\{\mu \})$
, and so
$\pi (\beta ) \notin G_\mu $
. Thus,
$\beta \in \mathcal {C}_3$
if and only if
$\pi (\beta )\in G_\mu $
, and so
$G_\mu $
is
$\boldsymbol {\Pi }^0_3$
-hard (and thus
$\boldsymbol {\Pi }^0_3$
-complete by Fact 5.2).
To prove that
$G^V$
is
$\boldsymbol {\Pi }^0_2$
-complete, first recall that it is a
$\boldsymbol {\Pi }^0_2$
set (Fact 5.5). Again by Lemma 7.2(ii), we see that
$G^V$
is dense. Given
$\nu \notin V$
, we know that
$G_\nu $
is dense. Note that
$G_\nu \subseteq X \setminus G^V$
, and thus both
$G^V$
and its complement are dense. It follows that
$G^V$
is
$\boldsymbol {\Pi }^0_2$
-complete (indeed, suppose that
$G^V$
is
$\boldsymbol {\Sigma }^0_2$
-complete; then both
$G^V$
and its complement would be dense
$\boldsymbol {\Pi }^0_2$
sets, which is impossible by the Baire category theorem). This completes the proof of Theorem 7.4.
Combined with the upper bounds of Facts 5.9 and 5.10, Theorem 7.4 has the following immediate corollary.
Corollary 7.5. If
$(X,T)$
is a compact dynamical system with the strong approximate product structure and
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is a non-empty closed connected set whose complement
$U={\mathcal {M}_{\mathit T}}(X)\setminus V$
is also non-empty, then:
-
(i) the sets
$G_V$
and
$G(V)$
are
$\boldsymbol {\Pi }_3^0$
-complete; -
(ii) the set
is
$\boldsymbol {\Sigma }_3^0$
-complete.
is 
Observe that for any ergodic probability measure
$\mu $
, it follows from the Birkhoff ergodic theorem that for
$\mu $
-almost all x, we have
$x\in B_\varphi (\mu (\varphi ))$
.
Theorem 7.6. Let
$(X,T)$
be a Polish dynamical system with the strong approximate product structure. If
$\varphi \in \mathcal {C}_b(X)$
and
$\alpha \in \mathbb {R}$
are such that
$$ \begin{align*} V_\varphi(\alpha)=\bigg\{\mu\in{\mathcal{M}_{\mathit T}}(X)\colon \int\varphi\,d\mu=\alpha\bigg\}\neq\emptyset \end{align*} $$
and
$U={\mathcal {M}_{\mathit T}}(X)\setminus V_\varphi (\alpha )\neq \emptyset $
, then
$B_\varphi (\alpha )$
is a
$\boldsymbol {\Pi }^0_3$
-complete set and
$I_\varphi (X,T)$
is a
$\boldsymbol {\Sigma }^0_3$
-complete set.
Proof. Pick any measure
$\nu \in U$
. As in the proof of Theorem 7.4, we can pick a measure
$\bar {\mu } \in V_\varphi (\alpha )$
such that
$t \nu + (1-t) \bar {\mu } \in U$
for all
$0 < t \le 1$
. We select an arbitrary sequence
$n_k \nearrow \infty $
, with
$n_{k+1} - n_k \to \infty $
. Given a sequence
$\beta \in \omega ^ \omega $
, we define function
$\psi ^\beta \colon \omega \to [0, 1]$
by setting
$\psi ^\beta (n_{2k}) = 1/(\beta (k) + 1)$
,
$\psi ^\beta (n_{2k+1}) = 0$
and interpolating linearly on other arguments, that is,
$$ \begin{align} \psi^\beta (j) = \frac{j - n_k}{n_{k+1} - n_k} \psi^\beta(n_k) + \frac{n_{k+1} - j}{n_{k+1} - n_k} \psi^\beta(n_{k+1}) \quad \textrm{for } n_k < j < n_{k+1}. \end{align} $$
It is straightforward to check that:
-
(i)
$|\psi ^\beta (j) - \psi ^\beta (j+1)| \to 0$
as
$j \to \infty $
; -
(ii) if
$\beta \to \infty $
, then
$\psi ^\beta (j) \to 0$
; -
(iii) if not
$\beta \to \infty $
, then both
$0$
and
$1 / (\liminf \beta (j) + 1)$
are limit points of sequence
$\psi ^\beta (j)$
.
We define measures
$$ \begin{align*} \mu^\beta_j = \psi^\beta(j) \nu + (1 - \psi^\beta(j)) \bar{\mu}. \end{align*} $$
From the properties of
$\psi ^\beta $
listed above, it follows that:
-
(i)
$D(\mu ^\beta _j, \mu ^\beta _{j+1}) \to 0$
as
$j \to \infty $
, and thus we can invoke Lemma 7.2 to obtain a point
$y^\beta $
; -
(ii) if
$\beta \to \infty $
, then
$\mu _j^\beta \to \bar {\mu }$
and consequently
$y^\beta \in G_{\bar {\mu }} \subseteq B_\varphi (\alpha )$
; -
(iii) if not
$\beta \to \infty $
, then
$y^\beta \in I_\varphi (X,T)$
.
Note that the measures
$\mu ^\beta _1 \cdots \mu ^\beta _{n_{2k}}$
depend only on the values
$\beta (1) \cdots \beta (k)$
. This, together with part (ii) of Lemma 7.2, implies that the map
$\pi \colon \omega ^ \omega \to X$
given by
$\beta \mapsto y^\beta $
is continuous. By parts (ii) and (iii),
$\pi $
is a continuous reduction of
$\mathcal {C}_3$
to
$B_\varphi (\alpha )$
, and also a reduction of
$ \omega ^ \omega \setminus \mathcal {C}_3$
to
$I_\varphi (X,T)$
. Together with Fact 5.4, this finishes the proof.
From the proof above, we also obtain the following corollary.
Corollary 7.7. If
$(X,T)$
is a Polish dynamical system with the strong approximate product structure with respect to a compatible metric
$\rho $
and there are at least two T-invariant measures, then:
-
(i) the quasi-regular set
$Q(X,T)$
is
$\boldsymbol {\Pi }_3^0$
-hard, and thus
$Q(X,T)$
is
$\boldsymbol {\Pi }^0_3$
-complete; -
(ii) the irregular set
$I(X,T)$
is
$\boldsymbol {\Sigma }_3^0$
-hard, and thus
$I(X,T)$
is
$\boldsymbol {\Sigma }_3^0$
-complete.
Proof. Pick any two T-invariant measures
$\bar {\mu } \neq \nu $
and a function
$\varphi \in \mathcal {C}_b(X)$
such that
$\alpha = \int _X \varphi \, d \mu \neq \int _X \varphi \, d \nu $
. In the proof of Theorem 7.6, we defined a continuous map
$\pi \colon \omega ^ \omega \to X$
such that
$\pi (\mathcal {C}_3) \subseteq G_{\bar {\mu }}$
and
$\pi ( \omega ^ \omega \setminus \mathcal {C}_3) \subseteq I_\varphi (X,T)$
. Since
$G_{\bar {\mu }} \subseteq Q(X,T)$
and
$ I_\varphi (X,T) \subseteq I(X,T)$
, the map
$\pi $
reduces
$\mathcal {C}_3$
to
$Q(X,T)$
, and reduces
$ \omega ^ \omega \setminus \mathcal {C}_3$
to
$I(X,T)$
. In Fact 5.4, we have shown the matching upper bounds.
8 Compactness and higher level completeness
In Theorem 7.4, it was shown for a Polish dynamical system
$(X,T)$
satisfying a weak form of specification (the strong approximate product structure) and having at least two invariant measures that for every closed connected
$V \subseteq \mathcal {M}_T$
that is non-trivial (that is,
$V\neq \emptyset $
,
$V\neq \mathcal {M}_T$
), we have that
$G(V)$
is
$\boldsymbol {\Pi }^0_3$
-hard. When X is compact, Fact 5.10 shows that
$G(V)$
is a
$\boldsymbol {\Pi }^0_3$
set (this holds for any closed
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
) and so
$G(V)$
is a
$\boldsymbol {\Pi }^0_3$
-complete set in this case. Similar results hold for the sets
$G_V=\{ x \in X\colon V(x) \subseteq V\}$
(where
$V\subseteq {\mathcal {M}_{\mathit T}}(X)$
is closed) and 
(where
$U\subseteq {\mathcal {M}_{\mathit T}}(X)$
is open). When X is not compact, the natural computation shows that
$G(V)$
and
$G_V$
are
$\boldsymbol {\Pi }^1_1$
sets, and 
is a
$\boldsymbol {\Sigma }^1_1$
set (see Facts 5.7 and 5.8). Recall from Fact 5.5 that in this case, we still have that for any closed
$V\subseteq \mathcal {M}_T$
,
$G^V=\{ x \in X \colon V\subseteq V(x)\}$
is
$\boldsymbol {\Pi }^0_2$
. A natural question is whether the compactness of X is necessary to reduce the complexity of
$G_V$
,
$G(V)$
and 
to a Borel set. The next result shows that this is the case as, in general, the sets
$G(V)$
and
$G_V$
may be a
$\boldsymbol {\Pi }^1_1$
-complete set, while 
is a
$\boldsymbol {\Pi }^1_1$
-complete set. Thus, we see a strong dichotomy in the complexity of these sets between the compact and non-compact cases.
Recall that a tree
$\mathcal {T}$
on
$\omega $
is a set
$\mathcal {T}\subseteq \omega ^{<\omega }$
of finite length sequences from
$\omega $
that is closed under the operation of taking initial segments. A tree is called wellfounded if it does not contain an infinite branch, and illfounded if it does. Let
$\text {Tr}$
denote the set of all trees on
$\omega $
. We fix an enumeration
$t^{(1)},t^{(2)},\ldots $
of
$\omega ^{<\omega }$
such that
$|t^{(i)}|\le |t^{(j)}|$
for
$i\le j$
. Using this enumeration, we identify
$\omega ^{<\omega }$
with
$\omega $
and
$\text {Tr}$
with a closed subset of the compact Polish space
$2^\omega $
. Thus,
$\text {Tr}$
is a Polish space and
$\text {Wf}\subseteq \text {Tr}$
is a
$\boldsymbol {\Pi }^1_1$
-complete set in
$\text {Tr}$
(see [Reference Kechris19, Ch. 32.B]). Note that two trees in
$\text {Tr}$
are close if and only if there is n such that for each
$1\le i \le n$
, the finite word
$t^{(i)}$
either belongs to both trees or belongs to neither of them.
Theorem 8.1. There is a Polish dynamical system
$(X,T)$
, so
$T \colon X\to X$
is continuous, and a closed, connected, non-empty
$V \subseteq {\mathcal {M}_{\mathit T}}(X)$
such that
$G(V)$
is
$\boldsymbol {\Pi }^1_1$
-complete. In fact, we may take
$V=\{ \mu \}$
for some
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
and
$(X,T)$
to have the specification property. Furthermore, setting
$U=\{\nu \in {\mathcal {M}_{\mathit T}}(X):\nu \neq \mu \}$
, we obtain that
$G_V$
is
$\boldsymbol {\Pi }^1_1$
-complete and 
is
$\boldsymbol {\Sigma }^1_1$
-complete.
Remark 8.2. For
$V=\{\mu \}$
, where
$\mu \in {\mathcal {M}_{\mathit T}}(X)$
, the sets
$G_\mu $
and
$G(V)$
may not be equal. For x to be in
$G_\mu $
, that is, for x to be a
$\mu $
generic point, we must have that for all increasing sequences of positive integers
$(n_i)_{i\ge 1}$
, the empirical measures
$\mathcal {E}(x,n_i)$
converge to
$\mu $
. For x to be in
$G(V)$
, we need only have that for each such sequence
$n_i$
, if the empirical measures
$\mathcal {E}(x,n_i)$
converge to a measure, then that measure must be
$\mu $
. This distinction is important, since the set
$G_\mu $
is always
$\boldsymbol {\Pi }^0_3$
by Fact 5.2.
Proof of Theorem 8.1
Let
$X=\omega ^\omega $
with the (continuous) shift map T. For
$n\in \omega $
, write
$\delta _{\bar n}$
for the point-mass measure on the constant n sequence denoted
$\bar {n}$
. Let
$V=\{ \delta _{\bar 0}\}$
.
To show that
$G(V)$
is
$\boldsymbol {\Pi }^1_1$
-complete, we construct a continuous map
$\pi \colon \text {Tr}\to X$
reducing
$\text {Wf}$
to
$G(V)$
, where
$\text {Wf}$
is the set of all wellfounded trees on
$\omega $
.
First, we will associate to each finite word
$s\in \omega ^{<\omega }$
a measure
$\mu _s$
that will be a finite combination of Dirac point-mass measures. Let us fix some auxiliary notation. By
$p(n)$
, we will denote the nth prime (that is,
$p(0)=2$
,
$p(1)=3$
etc.). Fix a bijection
$\mathbf {n}\colon \omega \times \omega \to \omega $
. Let Q be the set of positive integers divisible by at least two different primes. Let
$q(n)$
denote the nth smallest element of Q. That is,
$q(0)=6$
,
$q(1)=10$
,
$q(2)=12$
etc. Given
$(n,i)\in \omega \times \omega $
, we set
$\nu _{n,i}$
to be the Dirac point-mass measure concentrated on the constant sequence
$\bar p=ppp\ldots \in \omega ^\omega $
, where
$p=p(\mathbf {n}(n,i))$
. For
$s=(s(0)\ldots s(\ell -1))\in \omega ^{<\omega }$
(so
$\ell =|s|$
is the length of s), we define a measure
$\mu _s$
as
$$ \begin{align*} \mu_s=\frac{1}{2^\ell}\delta_{\bar{1}}+\sum_{i<\ell} \frac{1}{2^{i+1}}\nu_{s(i),i}. \end{align*} $$
Suppose
$s_0, s_1, s_2,\dots $
is a sequence from
$\omega ^{<\omega }$
with
$\lim _i |s_i|=\infty $
and such that
$s_i$
converges to
$x=(x(0),x(1),\dots ) \in \omega ^\omega $
, that is,
$\text { for all } m,\ \text { there exists } n\ \text { for all } i\geq n\ (s_i\restriction m=x\restriction m)$
. Then, the corresponding sequence of measures
$\mu _{s_i}$
is easily seen to converge to the measure
$\mu _{x}$
as
$i\to \infty $
, where
$\mu _x$
is the measure given by
$$ \begin{align*} \mu_{x}=\sum_{i=0}^\infty\frac{1}{2^{i+1}}\nu_{x(i),i}= \sum_{i=0}^\infty\frac{1}{2^{i+1}}\nu_{p(\mathbf{n}(x(i),i))}. \end{align*} $$
However, assume that
$s_0, s_1, s_2,\ldots $
is a sequence of distinct elements of
$\omega ^{<\omega }$
and the corresponding sequence of measures
$\mu _{s_i}$
is Cauchy. We claim that
$\lim _i |s_i|=\infty $
and there is an
$x \in \omega ^\omega $
such that
$s_i$
converges to x. Let k be the least integer, if one exists, such that either there are infinitely many i with
$s_i(k)$
not defined or
$\{ s_i(k) \}$
is not eventually constant. Assuming k is defined, the first case cannot happen as then,
$s_i\restriction k$
is defined and constant for all large enough i, but for some
$i<j$
, we would have
$s_i=s_j$
. If the second case in the definition of k occurs, then for large enough i, we have that
$s_i\restriction k$
is constant and either
$s_i(k)$
takes infinitely many values or there are two distinct values of
$s_i(k)$
both of which occur for infinitely many values of i. However, we would then have
$D(\mu _{s_i}, \mu _{s_j})\geq 1/2^{k+1}$
for arbitrarily large
$i<j$
, which is a contradiction. So, k is not defined, that is, for all k, we have that for all large enough i,
$s_i(k)$
is defined and constant. This shows that the
$s_i$
converge to some
$x \in \omega ^\omega $
.
Finally, for each
$s=(s(0)\ldots s(\ell -1))\in \omega ^{<\omega }$
and
$0\le i<\ell $
, we set
$n^s_i=p(\mathbf {n}(s(i),i))$
and define a periodic point
$$ \begin{align*} y^s= \Bigg( \underbrace{n^s_0\ldots n^s_0}_{2^{2\ell}\text{ times}}\ \underbrace{n^s_1\ldots n^s_1}_{2^{2\ell-1}\text{ times}}\ \ldots\ \underbrace{n^s_{\ell-1}\ldots n^s_{\ell-1}}_{2^{\ell+2}\text{ times}}\ \underbrace{1\ldots 1\vphantom{n^s_1}}_{2^{\ell+1}\text{ times}} \Bigg)^\infty\in\omega^\omega. \end{align*} $$
Note that the primary period of
$y^s$
equals
$p_s=2^{2\ell +1} - 2^{\ell + 1}$
and
$D(\mathcal {E}(y^s, p_s),\mu _s)$
goes to
$0$
as
$\ell $
goes to infinity. We are ready to define
$\pi (\mathcal {T})$
for a tree
$\mathcal {T}$
. Let
$c_n$
be the primary period of
$y^s$
for
$s=t^{(n)}$
. We fix sequences
$(a_n)$
and
$(b_n)$
satisfying
$a_1=2$
,
$a_{2n}=c_n$
,
$a_{2n+1}=1$
,
$b_{n+1}> b_n$
and
$$ \begin{align} a_nb_n> \max\{ 2^n a_{n+1}, 2^n (a_0 b_0+\cdots+a_{n-1}b_{n-1})\} \end{align} $$
for all n.
We set
$\ell _0=0$
. As stage
$n\ge 1$
of the construction of
$x=\pi (\mathcal {T})$
, we will define
$x_{[\ell _{n-1},\ell _n)}$
, where
$\ell _n=\ell _{n-1}+ a_n b_n$
. For n odd, we set
$x_{[\ell _{n-1},\ell _n)}={\bar {0}}_{[\ell _{n-1},\ell _n)}=0\ldots 0$
(here,
$0$
appears
$a_nb_n$
times). For n even, we have two cases: if
$s=t^{(n/2)}\in \mathcal {T}$
, then we put
$x_{[\ell _{n-1},\ell _n)}=(y^s)^{b_n}$
. Otherwise (that is, if
$s=t^{(n/2)}\notin \mathcal {T}$
), we put
$x_{[\ell _{n-1},\ell _n)}=(q_{n/2})^{a_nb_n}$
. In other words, for
$n\ge 1$
, we have
$$ \begin{align*} x_{[\ell_{n-1},\ell_{n})}= \begin{cases} 0^{a_nb_n} & \textrm{if } n \text{ is odd}, \\ (q_{n/2})^{a_{n}b_{n}} & \textrm{where } q_{n/2}=q(n/2) \text{ if } n \text{ is even and } s=t^{(n/2)}\notin\mathcal{T}, \\ (y^s_{[0, a_{n})})^{b_{n}} & \textrm{if } n \text{ is even and } s = t^{(n/2)}\in \mathcal{T}. \end{cases} \end{align*} $$
Note that
$\pi (\mathcal {T})=x$
defined in this way depends continuously on
$\mathcal {T}$
because
$x_{[0,\ell _{2n})}$
is determined by
$\{t^{(1)},\ldots ,t^{(n)}\}\cap \mathcal {T}$
. Furthermore, it follows from equation (25) that the measures
$\mathcal {E}(x, \ell _{2n-1})$
converge to
$\delta _{\bar 0}$
as
$n\to \infty $
. Thus,
$\delta _{\bar 0} \in V(x)$
. We show that
$\delta _{\bar 0}$
is the only measure in
$V(x)$
if and only if
$\mathcal {T} \in \text {Tr}$
is a wellfounded tree. Let
$\nu \in V(x)$
and
$i_k$
be a strictly increasing sequence with
$\lim _{k\to \infty } \mathcal {E}(x, i_k)=\nu $
. Let
$(n_k)$
be a sequence of integers such that
$\ell _{n_k} \leq i_k < \ell _{n_{k}+1}$
for every k. Without loss of generality, we assume that the
$(n_k)$
is strictly increasing and
$n_k$
is even for every k. The proof in the other case is similar.
Let
$e_k=(i_k-\ell _{n_k})/i_k$
. By passing to a subsequence, we may assume that
$e_k\to e\in [0,1]$
as
$k\to \infty $
. From equation (25), we have that the sequence
$\mathcal {E}(x, i_k)$
and the sequence with the kth term given by
$$ \begin{align*}\nu_k=(1-e) \mathcal{E}(x_{[\ell_{n_{k}-1},\ell_{n_k})})+ e\mathcal{E}(x_{[\ell_{n_{k}},\ell_{n_k+1})})\end{align*} $$
have the same weak
${}^*$
limit in
$\mathcal {M}_T(X)$
. If
$e=1$
, then
$\nu =\delta _{\bar {0}}$
, so assume
$e<1$
. Take the sequence
$(m_k)$
so that
$n_k=2m_k$
for every k.
Note that
$x_{[\ell _{n_{k}},\ell _{n_k+1})}$
is a sequence containing only zeros. Also,
$$ \begin{align} x_{[\ell_{n_{k}-1},\ell_{n_k})}= \begin{cases} q_k^{a_{n_k}b_{n_k}} & \textrm{where } q_k=q(m_k)\textrm{ if } t^{(m_k)}\notin\mathcal{T}, \\ (y^s_{[0, a_{n_k})})^{b_{n_k}} & \textrm{where } s = t(m_k) \textrm{ if } t^{(m_k)} \in \mathcal{T}. \end{cases} \end{align} $$
It follows that the sequence
$\nu _k$
converges to the same weak* limit as the sequence of measures
$$ \begin{align} w_k = \begin{cases} (1 - e) \delta_{\overline{q_k}} + e \delta_{\bar{0}} & \textrm{ if } t^{(m_k)}\notin\mathcal{T}, \\ (1 - e) \mu_{t^{(m_k)}} + e \delta_{\bar{0}} & \textrm{ if } t^{(m_k)} \in \mathcal{T}. \end{cases} \end{align} $$
Suppose
$t^{(m_k)}\notin \mathcal {T}$
for some subsequence
$(k(j))_{j \ge 1}$
. Then, the sequence of measures
$\delta _{\overline {q_{k(j)}}}$
would also converge, which is a contradiction. Thus, for sufficiently big k, the second case of equation (27) holds. It follows that the sequence
$(\mu _{t^{(m_k)}})$
converges. This, by our previous observations, gives that the
$t^{(m_k)}$
converge to some
$z \in \omega ^\omega $
. Since each
$t^{(m_k)} \in \mathcal {T}$
, this show that z is a branch through
$\mathcal {T}$
and so
$\mathcal {T}$
is illfounded. We have shown that if
$V(\pi (\mathcal {T})) \neq \{ \delta _{\bar 0}\}$
, then
$\mathcal {T}$
is illfounded. Conversely, if
$\mathcal {T}$
is illfounded, say z is a branch through
$\mathcal {T}$
, then along the sequence
$(n_{2k})$
, where
$t^{(k)}=z\restriction k$
, we have that
$\mathcal {E}(\pi (\mathcal {T}), n_{2k})$
converges to the measure
$\mu _z$
defined above and
$\mu _z \neq \delta _{\bar 0}$
, and so we have
$V(\pi (\mathcal {T})) \neq \{ \delta _{\bar 0}\}$
. So,
$V(\pi (\mathcal {T}))=\{\delta _{\bar {0}}\}$
if and only if
$\mathcal {T}\in \text {Wf}$
. This finishes the proof. It is easy to see that the above proof shows in fact that
$G_{\{\delta _{\bar {0}}\}}$
is
$\boldsymbol {\Pi }^1_1$
-complete, while 
, where
$U={\mathcal {M}_{\mathit T}}(X)\setminus \{\delta _{\bar {0}}\}$
, is
$\boldsymbol {\Sigma }^1_1$
complete.
9 Complexity of
$G_{\mu }$
in a minimal subshift
Recall that a dynamical system
$(X, T)$
is called minimal if
$\overline { \{T^n x \colon n \ge 0 \} } = X$
for every
$x \in X$
. Minimality is a stronger condition than transitivity. We give an example that shows that
$G_\mu $
may be
$\boldsymbol {\Pi }_3^0$
-complete in a minimal system.
Theorem 9.1. There exists a minimal shift
$(X, T)$
and an ergodic measure
$\mu $
such that
$G_\mu $
is
$\boldsymbol {\Pi }_3^0$
-complete.
We will examine a construction given by Oxtoby, originally used as an example of a minimal, not uniquely ergodic dynamical system; see [Reference Downarowicz, Kolyada, Manin and Ward11, Example 10.3] or [Reference Oxtoby29, §10]. We will construct a subshift of
$\{0, 1\}^\omega $
. The construction uses a sequence
$(s_j)_{j \geq 1}$
of natural numbers such that
$$ \begin{align} \sum_{j=1}^{\infty} \frac{1}{s_j} < 1. \end{align} $$
We will now define certain strings over the alphabet
$\{0, 1, \texttt {x}\}$
. Here,
$\texttt {x}$
is a placeholder to be filled at a later stage of the construction. We begin with
To illustrate, for
$s_1=3$
,
$s_2=4$
and
$s_3=3$
, the sequence
$(W_n)$
begins with
$$ \begin{align*} W_0 &= \texttt{x}, \\ W_1 &= \texttt{0xx}, \\ W_2 &= \texttt{0110xx0xx0xx}, \\ W_3 &= \texttt{0110000000000110xx0xx0xx0110xx0xx0xx}. \end{align*} $$
For all n, the word 
can be obtained by replacing all
$\texttt {x}$
symbols with zeros in
$W_n$
. Similarly, the word 
can be obtained by replacing all
$\texttt {x}$
symbols with ones in
$W_n$
. The lengths of 
and 
are equal, we will denote this length by
$l_n$
. Observe that in the sequence 
, each word is a prefix of the next one and thus, we can define
$y \in \{0, 1\}^\omega $
to be the limit of this sequence. Finally, we define X to be the closure of the orbit of y. The sequence y is Toeplitz and X is a minimal subshift; see [Reference Downarowicz, Kolyada, Manin and Ward11] for details. Note that the language of X can be described as follows:
For a word u, define
$c(u) = | \{ 0 \le i < |u| \colon u_i = 1\}|$
and
$m(u) = c(u) / |u|.$
From equation (29), it follows that 
is a weakly increasing sequence and, therefore, 
for some real number
$a \in [0, 1]$
. Similarly, 
for some
$b \in [0, 1]$
. Again from equation (29), we get 
. The condition (28) implies that
$\prod _{j=1}^\infty (1 - 1/s_j)> 0$
and, consequently,
$a < b$
.
Fact 9.2. The set
${\mathcal {M}_{\mathit T}^e}(X)$
consists of two measures
$\mu _0, \mu _1$
. Moreover,
$\mu _0(C) = a$
and
$\mu _1(C) = b$
, where
$C = \{ x \in X : x_0 = 1 \}$
.
Proof. The shift space X has exactly two ergodic measures [Reference Downarowicz, Kolyada, Manin and Ward11, Ch. 14]. Note that for arbitrary continuous
$f \colon X \to \mathbb {R}$
, we have
$$ \begin{align} \min_{\mu \in {\mathcal{M}_{\mathit T}^e}(X)} \int_X f \,d \mu = \min_{\mu \in {\mathcal{M}_{\mathit T}}(X)} \int_X f \,d \mu = \lim_{k \to \infty} \min_{x \in X} A_k f(x). \end{align} $$
We apply equation (30) to the characteristic function
$\chi _C$
. We claim that the right-hand side of equation (30) is then equal to a. Indeed, note that in this particular case, the right-hand side simplifies to
$$ \begin{align} \lim_{k \to \infty} \min_{u \in L_k(X)} m(u), \end{align} $$
where
$L_k(X) = \{ u \in L(X) \colon |u| = k \}$
. Pick any
$n \ge 0$
, then any word in
$L(X)$
can be divided into blocks, each equal to 
or 
, where the first and last block may be incomplete. This shows that equation (31) is greater or equal to 
. Since this is true for any n, we get that equation (31) is greater or equal to a. However, by plugging 
, we find that equation (31) is less than or equal to a. Thus, there exists a measure
$\mu _0 \in {\mathcal {M}_{\mathit T}^e}(X)$
such that
$\mu _0(C) = a$
. Similarly, applying equation (30) to
$- \chi _C$
shows us that there exists a measure
$\mu _1 \in {\mathcal {M}_{\mathit T}^e}(X)$
such that
$\mu _1(C) = b$
.
Fact 9.3. For any k, suppose we have
$j_0, j_1, \dots j_k$
such that
$0 \leq j_i \leq s_{i+1} - 2$
. Then, the word 
is a suffix of 
, in particular, it belongs to the language L.
Proof. Let us write
$u \sqsubseteq v$
whenever the word u is a suffix of v. We will proceed by induction. For
$k=0$
, the statement is true. Now, suppose the statement holds for
$k-1$
, then we have 
. From this, we obtain
$ 0_0^{j_0} 0_1^{j_1} \cdots 0_{k-1}^{j_{k-1}} 0_{k}^{j_{k}} \sqsubseteq 0_k^{j_k + 1} \sqsubseteq 0_k^{s_{k+1} - 1} \sqsubseteq 0_{k+1}. $
Since the relation
$\sqsubseteq $
is transitive, we are done.
From the fact above, it follows that the infinite sequence
$0_0^{j_0} 0_1^{j_1} \cdots 0_k^{j_k} \cdots $
is an element of X if
$1 \leq j_i \leq s_{i+1} - 2$
for all i. Recall that the set
$ \mathcal {C}_3 := \{ \beta \in \omega ^ \omega : \liminf \beta (n) = +\infty \} $
is a
$\boldsymbol {\Pi }_3^0$
-complete subset of
$\in \omega ^ \omega $
. We define
$ f : \omega ^ \omega \rightarrow X $
by the formula
We claim that
$\beta \in \mathcal {C}_3 \Leftrightarrow f(\beta ) \in G_{\mu _0}$
. Fact 9.2 implies that
$G_{\mu _0} = B_{\chi _C}(a)$
. This will simplify our analysis considerably.
-
. Let
$k = \liminf \beta (i) + 1$
. Consider any i such that
$j_i = k$
. From equation (28), note that is a prefix of . Thus, is a prefix of
$f(\beta )$
. The underlined part is a suffix of and, hence, . We have and . We treat the underlined part as if it contained only zeros; this gives the estimate as
$i \to \infty $
. Thus,
$f(\beta ) \not \in B_{\chi _C}(a)$
.
-
. Any prefix of
$f(\beta )$
can be written as (32)where
$k \ge 0$
and u is a prefix of . Recall that and . So we further have either , where
$u'$
is a prefix of , or , where
$u'$
is a prefix of . In both cases, we have
$|u'| \le l_{2i-1}$
and, consequently, As in the previous case, the underlined part of equation (32) has length
$\le l_{2i-2}$
. Thus, we obtain Finally, since
$|w| \ge j_{i-1} l_{2i-1}$
and for all n, we get Our assumption
$$ \begin{align*} m(w) \le \frac{3}{j_{i-1}} + a. \end{align*} $$
$\beta \in \mathcal {C}_3$
implies that
$j_i \to \infty $
and, hence,
$\limsup m(w) \le a$
as the length of w goes to infinity. Fact 9.2 implies that
$\liminf m(w) \ge a$
. Since w is an arbitrary prefix of
$f(\beta )$
, we get that
$f(\beta ) \in B_{\chi _C}(a) = G_{\mu _0}$
.
is a prefix of 
. Thus, 
and, hence, 
. We have 
and 
. We treat the underlined part as if it contained only zeros; this gives the estimate 
. Recall that 
and 
. So we further have either 
, where 
, or 
, where 
. In both cases, we have 
for all n, we get 
Thus, f is a reduction of
$\mathcal {C}_3$
to
$G_{\mu _0}$
, which proves that the latter set is
$\boldsymbol {\Pi }_3^0$
-complete. We have completed the proof of Theorem 9.1.
We claim that the minimal system X constructed in this section does not have SAPS. Consider any word
$w \in L(X)$
of length
$l_n$
. It follows from the construction that w is a subword of one of the words 
, 
, 
, 
. Consequently, we obtain the bound
$|L_{l_n}(X)| \le 4l_n + 4$
. Thus,
$$ \begin{align*} \liminf_{n \to \infty} \frac{|L_n(X)|}{n} \le 4. \end{align*} $$
By a result of Cyr and Kra [Reference Cyr and Kra9, Theorem 1.1], this implies that there are at most four distinct measures in
${\mathcal {M}_{\mathit T}}(X)$
whose set of generic points is non-empty. Thus, for a suitable choice of
$t \in [0, 1]$
, the measure
$\nu = t\mu _1 + (1-t) \mu _0$
will have
$G_\nu = \emptyset $
. However, if X had SAPS, by Corollary 7.3, the set
$G_\nu $
would be dense, in particular, non-empty.
Acknowledgments
Konrad Deka and Dominik Kwietniak were supported by the National Science Centre, Poland, grant Preludium-bis no. 2019/35/O/ST1/02266. Steve Jackson was supported by NSF grant 1800323. Bill Mance is supported by the National Science Centre, Poland, grant Sonata-bis no. 2019/34/E/ST1/00082 for the project. We would like to thank the referee for careful reading and many remarks that helped us to improve this paper.
