1 Introduction
Measuring the complexity of relations on structures is a very general task. In this paper we use the notion of Borel reducibility (see Definition 2.1) and the results of invariant descriptive set theory to compare the complexities of classification problems. For more details on invariant descriptive set theory we refer to the book by Gao [Reference Gao8]. For a short and nice introduction to the theory of Borel reductions we refer to a paper by Foreman [Reference Foreman7].
Several equivalence relations became milestones in this theory. Let us mention four of those, which describe an increasing chain of complexities:
-
• the equality on an uncountable Polish space,
-
• the equality of countable sets of real numbers,
-
• the
$S_{\infty }$
-universal orbit equivalence relation (
$S_{\infty }$
is the group of permutations on
$\mathbb {N}$
), -
• the universal orbit equivalence relation.
Let us give several examples to make the reader more familiar with the above relations. A classical example is a result of Gromov (see, e.g., [Reference Gao8, Theorem 14.2.1]) who proved that the isometry equivalence relation of compact metric spaces is a smooth equivalence relation, which means that it is Borel reducible to the equality of real numbers (or equivalently of an uncountable Polish space). The isomorphism relation of countable graphs or the isomorphism relation of countable linear orders are Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence. The homeomorphism equivalence relation of compact metrizable spaces and the isometry relation of separable complete metric spaces were proved by Zielinski in [Reference Zielinski29] and by Melleray in [Reference Melleray21], respectively, to be Borel bireducible to the universal orbit equivalence relation (see the survey paper by Motto Ros [Reference van Mill23]).
In order to capture all the structures in one space we need some sort of coding. This can be done by considering some universal space (e.g., the Hilbert cube or the Urysohn space) and all its subspaces with some natural Polish topology or Borel structure (e.g., the hyperspace topology or the Effros Borel structure). Sometimes there are other natural ways to encode a given structure. For example the class of separable complete metric spaces can be coded by the set of all metrics on
$\mathbb {N}$
where two metrics are defined to be equivalent if the completions of the respective spaces are isometric. Fortunately in this case, by [Reference Gao8, Theorem 14.1.3] it does not matter which of the two coding we choose. It is generally believed that this independence on a natural coding is common to other structures and thus the statements are usually formulated for all structures without mentioning the current coding. Nevertheless, for the formal treatment some coding is always necessary.
The aim of this paper is to determine the complexity of some classification problems of dynamical systems up to conjugacy. Dynamical systems of a fixed compact metrizable space X can be naturally coded as a space of continuous mappings of X into itself, with the uniform topology. This one as well as the subspace of all self-homeomorphisms is well known to be a Polish space.
Let us mention several results which are dealing with the complexity of conjugacy equivalence relation. It was proved by Hjorth that conjugacy equivalence relation of homeomorphisms of [0,1] is classifiable by countable structures [Reference Hjorth12, Section 4.2] (in fact Borel bireducible to the universal
$S_{\infty }$
-orbit equivalence relation) but conjugacy of homeomorphisms of
$[0,1]^2$
is not [Reference Hjorth12, Section 4.3]. By a result of Camerlo and Gao, conjugacy equivalence relation of both selfmaps and homeomorphisms of the Cantor set are Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation [Reference Camerlo and Gao4, Theorem 5]. Kaya proved that conjugacy of pointed minimal Cantor dynamical systems is Borel bireducible to the equality of countable subsets of reals [Reference Kaya14]. Conjugacy of odometers is smooth due to Buescu and Stewart [Reference Buescu and Stewart2]. The complexity of conjugacy of Toeplitz subshifts was treated several times—by Thomas, Sabok, and Tsankov, and by Kaya [Reference Kaya13, Reference Sabok and Tsankov26, Reference Sher28]. Conjugacy of two-sided subshifts is Borel bireducible to the universal countable Borel equivalence relation due to Clemens [Reference Clemens6]. There is an extensive exposition of results on the complexity of conjugacy equivalence relation on subshifts of
$2^G$
for a countable group G in the book by Gao, Jackson, and Seward [Reference Gao, Jackson and Seward9, Chapter 9]. Recently, during the 8th Visegrad Conference on Dynamical Systems in 2019 it was announced by Dominik Kwietniak that conjugacy of shifts with specification is Borel bireducible to the universal countable Borel equivalence relation.
In this paper, we deal with some of the missing parts. By mainly elementary and standard tools (excluding the complexity level of countable structures), we prove:
Theorem A (See Theorem 3.16).
The conjugacy equivalence relation of interval maps is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation.
Also, we prove:
Theorem B (See Theorem 4.10).
The conjugacy equivalence relation of homeomorphisms as well as conjugacy of selfmaps of the Hilbert cube is Borel bireducible to the universal orbit equivalence relation.
To this end we use some tools of infinite dimensional topology and a result of Zielinski on the complexity of homeomorphism equivalence relation of metrizable compacta [Reference Zielinski29] combining with some ideas of Krupski and the second author [Reference Krupski and Vejnar19]. Finally we make a small overview on the complexity of conjugacy equivalence relation of dynamical systems on the Cantor set, on the interval, on the circle, and on the Hilbert cube.
2 Definitions and notations
Let us define some standard notions from descriptive set theory (see, e.g., [Reference Kechris16]). A Polish space is a separable completely metrizable topological space. Recall that a standard Borel space is a measurable space
$(X, \mathcal S)$
such that there is a Polish topology
$\tau $
on X for which the family of Borel subsets of
$(X,\tau )$
is equal to
$\mathcal S$
. In order to compare the complexities of equivalence relations we use the notion of Borel reducibility.
Definition 2.1. Suppose that X and Y are sets and let E, F be equivalence relations on X and Y respectively. We say that E is reducible to F, and we denote this by
$E\leq F$
, if there exists a mapping
$f\colon X\to Y$
such that
$$\begin{align*}x \mathbin{E} x' \iff f(x) \mathbin{F} f(x'),\end{align*}$$
for every
$x, x'\in X$
. The mapping f is called a reduction of E into F. If the sets X and Y are endowed with Polish topologies (or standard Borel structures), we say that E is Borel reducible to F, and we write
$E\leq _B F$
, if there is a reduction
$f\colon X\to Y$
of E into F which is Borel measurable. We say that E is Borel bireducible to F, and we write
$E \sim _B F$
, if E is Borel reducible to F and F is Borel reducible to E.
In a similar fashion we define being continuously reducible if in addition X and Y are Polish spaces and f is continuous.
In the whole paper we set
$\mathbb N$
for all positive integers,
$I=[0,1]$
and we denote the closure operator by
$\text {Cl}$
. For a separable metric space X we denote by
$\mathcal K(X)$
the hyperspace of all compacta in X with the Hausdorff distance
$d_H$
and the corresponding Vietoris topology. If X is a Polish space
$\mathcal K(X)$
is known to be Polish. For compact metric spaces
$X, Y$
we consider the space
$C(X, Y)$
of all continuous mappings of X into Y with the supremum metric. In this way we get a Polish space. Especially the collection of all continuous selfmaps of X is denoted shortly by
$C(X)$
. We also denote by
$\text {Inj}(X, Y)$
the collection of all embeddings of X into Y and by
$\mathcal H(X)$
the collection of all homeomorphisms of X. These are again known to be Polish spaces.
The equality equivalence relation of real numbers is denoted by
$E_=$
. We denote by
$E_{=^+}$
the equivalence relation on
$\mathbb {R}^{\mathbb {N}}$
defined by
$(a_n)E_{=^+} (b_n)$
if and only if
$\{a_n\colon n\in \mathbb {N}\}=\{b_n\colon n\in \mathbb {N}\}$
. The last equivalence relation is called the equality of countable sets.
We say that an equivalence relation E defined on a standard Borel space X is classifiable by countable structures if there is a countable relational language
$\mathcal L$
such that E is Borel reducible to the isomorphism relation of
$\mathcal L$
-structures whose underlying set is
$\mathbb {N}$
. An equivalence relation E on a standard Borel space X is said to be an orbit equivalence relation if there is a Borel action of a Polish group G on X such that
$xEx'$
if and only if there is some
$g\in G$
for which
$gx=x'$
.
Let
${\mathcal C}$
be a class of equivalence relations on standard Borel spaces. An element
$E\in {\mathcal C}$
is called universal for
${\mathcal C}$
if
$F\leq _B E$
for every
$F\in {\mathcal C}$
. It is known that for every Polish group G there is an equivalence relation (denoted by
$E_G$
) on a standard Borel space that is universal for all orbit equivalence relations given by continuous G-actions. We are particularly interested in the universal
$S_{\infty }$
-equivalence relation
$E_{S_{\infty }}$
, where
$S_{\infty }$
is the group of all permutations of
$\mathbb {N}$
. It is known that an equivalence relation is classifiable by countable structures if and only if it is Borel reducible to
$E_{S_{\infty }}$
. Moreover
$E_{S_{\infty }}$
is known to be Borel bireducible to isomorphism equivalence relation of countable graphs. Also there exists a universal orbit equivalence relation which is denoted by
$E_{G_{\infty }}$
. We should also note that all the mentioned equivalence relations are analytic sets, i.e., images of standard Borel spaces with respect to a Borel measurable map. We have a chain of complexities
$$\begin{align*}E_=\leq_B E_{=^+}\leq_B E_{S_{\infty}}\leq_B E_{G_{\infty}} \end{align*}$$
and it is known that none of these Borel reductions can be reversed.
3 Interval dynamical systems
In this section we prove that conjugacy of interval dynamical systems is classifiable by countable structures. The strategy of our proof is as follows. In the first part we describe a natural reduction of interval dynamical systems to some kinds of countable structures. We assign to every
$f\in C(I)$
a countable invariant set
$C_f\subseteq I$
of some dynamically exceptional points for f. Since the set
$C_f$
does not need to be dense in I we do not have enough information to capture the dynamics of f by restricting to
$C_f$
. On the other hand the dynamics on the maximal open intervals of
$I\setminus C_f$
is quite simple. Hence it will be enough to define an invariant countable dense subset
$D_f$
in
$I\setminus \text {Cl}(C_f)$
arbitrarily. Consequently, we get that for f conjugate to g there exists a conjugacy of f to g which sends the set
$C_f\cup D_f$
onto
$C_g\cup D_g$
. Finally it is enough to assign to every
$f\in C(I)$
a countable structure
$\Psi (f)$
whose underlying set is
$C_f\cup D_f$
and which is equipped with one binary relation
$\leq \restriction _{C_f\cup D_f}$
and one mapping
$f\restriction _{C_f\cup D_f}$
(which can be as usual considered as a binary relation). We will prove then that if two such structures
$\Psi (f)$
and
$\Psi (g)$
are isomorphic then f and g are conjugate.
In the second part we prove that this reduction can be modified using some sort of coding so that the assigned countable structures share the same support and so that the new reduction is Borel. To this end we use the Lusin–Novikov selection theorem [Reference Kechris16, Theorem 18.10] several times.
For
$g\in C(I)$
we denote by
$\text {Fix}(g)$
the set of fixed points of g, i.e., those points for which
$g(x)=x$
. We omit the proof of the following “folklore” lemma. The key idea of the proof is the back and forth argument.
Lemma 3.1. Let
$f, g\in C(I)$
be increasing homeomorphisms such that
$\text {Fix}(f)=\text {Fix}(g)=\{0,1\}$
and let
$A, B\subseteq (0,1)$
be countable dense sets that are invariant in both directions for f and g respectively. Then there is a conjugacy h of f and g satisfying
$h(A)=B$
.
Definition 3.2. For
$f\in C(I)$
let us say that a point
$z\in I$
is a left sharp local maximum of f if there is some
$\delta>0$
such that
$f(x)<f(z)$
for
$x\in (z-\delta , z)$
and
$f(x)\leq f(z)$
for
$x\in (z, z+\delta )$
. In a similar fashion we define left sharp local minimum, right sharp local minimum, and right sharp local maximum.
Notation 3.3. Let
$M_f$
be the union of
$\{0,1\}$
and the set of all left and right sharp local maxima and minima. It is easily shown that the set
$M_f$
is countable. For a closed set
$F\subseteq I$
denote by
$\text {Acc}(F)$
the set of all accessible points of F in
$\mathbb R$
, i.e., those points
$x\in F$
for which there exists an open interval
$(a,b)\subseteq \mathbb R\setminus F$
for which
$x=a$
or
$x=b$
.
For every
$f\in C(I)$
let us denote by
$C_f$
the smallest set such that:
-
(a)
$M_f\subseteq C_f$
, -
(b) if
$f^{-1}(y)$
contains an interval then
$y\in C_f$
, -
(c) if
$n\in \mathbb N$
then
$\text {Acc}(\text {Fix}(f^n))\subseteq C_f$
, -
(d)
$f(C_f)\subseteq C_f$
, -
(e) if
$y\in C_f$
then
$\text {Acc}(f^{-1}(y))\subseteq C_f$
.
Lemma 3.4. The set
$C_f$
is countable for every
$f\in C(I)$
.
Proof Let
$S_1$
be the union of
$M_f$
, all the values of f at locally constant points and all the sets
$\text {Acc}(\text {Fix}(f^n))$
for
$n\in \mathbb N$
. Clearly
$S_1$
is countable. Let
$S_{i+1}=S_i\cup f(S_i)\cup \bigcup \{\text {Acc}(f^{-1}(y))\colon y \in S_i\}$
. Clearly
$C_f=\bigcup \{S_i\colon i\in \mathbb {N}\}$
and thus it is countable.
Note that
$C_f$
depends only on the topological properties of I and the dynamics of f. That is if f and g are conjugate by some homeomorphism h, then
$h(C_f)=C_g$
. This is clear because h maps
$M_f$
onto
$M_g$
, locally constant intervals of f to locally constant intervals of g and periodic points of f to periodic points of g.
Let us denote by
$\mathcal J_f$
the collection of all maximal open subintervals of
$I\setminus C_f$
.
Lemma 3.5. Let
$J\in \mathcal J_f$
. Then either
$f\restriction _J$
is constant or
$f\restriction _J$
is one to one and in this case
$f(J)\in \mathcal J_f$
. Also
$f^{-1}(J)$
is the finite union (possibly the empty union) of elements of
$\mathcal J_f$
.
Proof Let us prove first that
$f\restriction _J$
is either constant or one-to-one. Suppose that the contrary holds. Then there are points
$x, y, z\in J$
such that
$f(x)=f(y)\neq f(z)$
and
$x\neq y$
. Let us suppose that
$x<y<z$
and
$f(x)<f(z)$
(the other possibilities are just easy modifications). Let
$u=\min f\restriction _{[x,z]}$
and let
$v=\max (f^{-1}(u)\cap [x, z])$
. It follows that
$v\in (x,z)$
is a right sharp local minimum. By Notation 3.3(a) it follows that
$v\in C_f$
which is a contradiction since J is disjoint from
$C_f$
.
Suppose now that
$f\restriction _J$
is one-to-one and let us prove that
$f(J)\in \mathcal J_f$
. Observe first that
$f(J)$
is disjoint from
$C_f$
; otherwise there would be a point
$y\in f(J)\cap C_f$
and since
$f^{-1}(y)$
is a closed set not containing the whole set J there will be a point in
$\text {Acc} f^{-1}(y)\cap J$
which is a contradiction with Notation 3.3(e). We need to prove that
$f(J)$
is a maximal interval disjoint from
$C_f$
. Suppose that
$J=(a,b)$
. Then there are
$a_n, b_n\in C_f$
such that
$a_n\to a, b_n\to b$
. By continuity of f it follows that
$f(a_n)\to f(a)$
and
$f(b_n)\to f(b)$
. Also
$f(a_n), f(b_n)\in C_f$
by Notation 3.3(d). Thus the maximality follows.
Observe first that
$f^{-1}(J)$
is a countable union of disjoint collection of open intervals and if we prove that each of the intervals is mapped by f onto J it will follow by continuity that such a collection is in fact finite. Denote
$(a,b)=J$
and let
$(c,d)$
be a maximal interval in
$f^{-1}(J)$
. Clearly
$(c,d)\cap C_f=\emptyset $
by Notation 3.3(d), so it is enough to prove that it is maximal with this property. Note that
$f(c), f(d)\in \{a, b\}$
; otherwise we get a contradiction with
$(c,d)$
being maximal interval in
$f^{-1}(J)$
. Also it cannot happen that
$f(c)=f(d)$
; otherwise there will be a point of left local maximum or minimum in
$(c,d)$
which would produce a point in
$M_f\cap (c,d)$
, which in turn would give a point in
$C_f\cap J$
, by Notation 3.3(a) and (d). Hence
$f((c,d))=J$
. Moreover, by the first part of this proof we get that
$f\restriction _{(c,d)}$
is one-to-one and thus it is either increasing or decreasing. Without loss of generality suppose the first case. Let us distinguish several possibilities. If
$f\geq f(c)$
on a left neighborhood of c then c is a point of right sharp local minimum and thus
$c\in C_f$
. Otherwise choose a sequence
$a_n\in C_f$
such that
$a_n\to a$
. We define points
$c_n=\max ([0,c]\cap f^{-1}(a_n))$
. These are eventually well defined,
$c_n\to c$
and
$c_n\in \text {Acc} f^{-1}(a_n)$
. Hence by Notation 3.3(e)
$c_n\in C_f$
. We can proceed in a similar way with the point d and thus the interval
$(c,d)$
is maximal subinterval of
$I\setminus C_f$
.
Example 3.6. For the tent map
$f(x) =\min \{ 2x, 2(1-x)\}$
, the set
$C_f$
contains all the dyadic numbers in I; thus
$C_f$
is a dense subset of I and hence
$\mathcal J_f=\emptyset $
. For the map
$g=\frac 14 f$
we have
$$ \begin{align*} C_g&=\{2^{-n}, 1-2^{-n}\colon n\in\mathbb N\}\cup\{0,1\}, \\ \mathcal J_g&=\{(2^{-n-1}, 2^{-n}), (1-2^{-n}, 1-2^{-n-1})\colon n\in\mathbb N\}. \end{align*} $$
Notation 3.7. Let
$G_f$
be a directed graph on
$\mathcal J_f$
where
$(J,K)$
forms an oriented edge if and only if
$f(J)=K$
. Note that for every
$K\in \mathcal J_f$
there are only finitely many
$J\in \mathcal J_f$
for which
$f(J)=K$
. Hence every vertex of the graph
$G_f$
admits only finitely many arrows to enter. Let
$E_f=\mathbb Q\cap I\setminus \text {Cl}(C_f)$
and let
$$\begin{align*}D_f=\bigcup_{n\in\mathbb Z} f^n(E_f). \end{align*}$$
Note that the union is taken over all integers. In spite of that it follows by Lemma 3.5 that
$D_f$
is countable. Let us define
$$\begin{align*}\Psi(f)=(C_f\cup D_f, \leq\restriction_{C_f\cup D_f}, f\restriction_{C_f\cup D_f}). \end{align*}$$
Theorem 3.8. The mapping
$\Psi $
is a reduction of orientation preserving conjugacy of interval dynamical systems to the isomorphism relation of countable structures.
Proof Suppose first that f is conjugate to g via some increasing homeomorphism h, that is
$f=h^{-1}gh$
. We want to find an isomorphism
$\varphi \colon \Psi (f) \to \Psi (g)$
. Since h does not need to map
$D_f$
to
$D_g$
, we need to do some more work. In fact we find a conjugacy
$\bar h$
of f and g such that
$\bar h(C_f\cup D_f)=C_g\cup D_g$
. Then it will be enough to define a mapping
$\varphi \colon C_f\cup D_f\to C_g\cup D_g$
as the restriction of
$\bar h$
. We will define
$\bar h$
by parts. First of all we define
$\bar h$
on the set
$\text {Cl}(C_f)$
in the same way as h.
Clearly h induces an isomorphism of the graphs
$(\mathcal J_f, G_f)$
and
$(\mathcal J_g, G_g)$
. We will consider the components of the symmetrized graphs
$G_f$
and
$G_g$
. Note that
$J, K\in \mathcal J_f$
are in the same component of
$G_f$
if there are
$m, n\geq 0$
such that
$f^m(J)=f^n(K)$
.
Let us distinguish two cases for the components of
$G_f$
. If a component of
$G_f$
contains an oriented cycle, choose an element J in there (note that the cycle is unique). Hence there is
$n\in \mathbb N$
such that
$f^n(J)=J$
. By using Notation 3.3(c) it follows that either all the points of J are fixed points for
$f^n$
or there are no fixed points of
$f^n$
in J and the same has to be true for
$g^n$
on
$h(J)$
. In the first case we just let
$\bar h\restriction J$
to be any homeomorphism of
$\text {Cl}(J)$
and
$\text {Cl}(h(J))$
, and in the second case we obtain by Lemma 3.1 that there is a conjugacy
$\bar h\restriction _{\text {Cl}(J)}$
of
$f^n\restriction _{\text {Cl}(J)}$
and
$g^n\restriction _{\text {Cl}(h(J))}$
sending
$D_f\cap J$
onto
$D_g\cap h(J)$
. In components that do not contain an oriented cycle we choose J arbitrarily, and let
$\bar h\restriction _J$
be an arbitrary increasing homeomorphism
$J\to h(J)$
which maps
$J\cap D_f$
onto
$h(J)\cap D_g$
.
For any K that is in the same component as J find
$m, n\geq 0$
such that
$f^m(J)=f^n(K)\in \mathcal J_f$
and define
$\bar h$
on K using the definition of
$\bar h$
on J as
$$\begin{align*}(g^{-n}\restriction_{h(K)})g^m\bar h (f^{-m}\restriction_J)f^n. \end{align*}$$
On the other hand suppose that
$\varphi $
is an isomorphism of the countable structure
$\Psi (f)$
to
$\Psi (g)$
. Hence
$\varphi \colon C_f\cup D_f\to C_g\cup D_g$
is a bijection preserving the order. Thus it can be extended to an increasing homeomorphism
$\tilde {\varphi }\colon I\to I$
. We claim that
$\tilde \varphi $
conjugates f and g. Consider any point
$x\in C_f\cup D_f$
and compute
$$\begin{align*}g(\tilde\varphi(x))=g(\varphi(x))=\varphi(f(x))=\tilde\varphi(f(x)), \end{align*}$$
where the middle equality follows from
$\varphi $
being an isomorphism of
$\Psi (f)$
and
$\Psi (g)$
. Since the set
$C_f\cup D_f$
is dense it follows by continuity that
$g(\tilde \varphi (x))= \tilde \varphi (f(x))$
for every
$x\in I$
. Hence f and g are conjugate.
3.1 Borel coding
We need to verify that the mapping
$\Psi $
that was proved in Theorem 3.8 to be a reduction can be coded in a Borel way. We use standard notation for the Borel hierarchy, especially
$\Sigma ^0_1$
is used for the collection of all open sets,
$\Sigma ^0_2$
is used for the collection of countable unions of closed sets, etc. For a set
$B\subseteq X\times Y$
and
$x\in X$
let us denote by
$B_x$
the set
$\{y\in Y\colon (x,y)\in B\}$
and call it vertical section of B.
The following seems to be folklore in descriptive set theory, but for the sake of completeness we include a proof.
Proposition 3.9. Let
$X, Y$
be Polish spaces and
$B\subseteq X\times Y$
be a Borel set with countable vertical sections. Then the set
$\bigcup _{x\in X} \{x\}\times \text {Cl}(B_x)$
is Borel as well.
Proof Let
$\mathcal B$
be a countable base for the topology of Y. By the Lusin–Novikov selection theorem, we can assume that
$B=\bigcup f_n$
for some Borel maps
$f_n\colon X\to Y$
. It follows that
$$\begin{align*}(X\times Y)\setminus \left(\bigcup_{x\in X} \{x\}\times \text{Cl}(B_x)\right)=\bigcup_{U\in \mathcal B}\bigcap_{n\in\mathbb N}((X\setminus f_n^{-1}(U))\times U). \end{align*}$$
Hence the set under discussion is Borel.
Let us denote
$\Gamma =\{(K, a)\in \mathcal K(I)\times I\colon a\in \text {Acc}(K)\}$
.
Lemma 3.10. The set
$\Gamma $
is a
$\Sigma ^0_2$
-set.
Proof The sets
$$\begin{align*}L_n=\{(K, a)\in \mathcal K(I)\times I\colon a\in K, K\cap (a-2^{-n}, a)=\emptyset\},\end{align*}$$
$$\begin{align*}R_n=\{(K, a)\in \mathcal K(I)\times I\colon a\in K, K\cap (a, a+2^{-n})=\emptyset\}\end{align*}$$
are closed for every
$n\in \mathbb N$
. Hence the set
$\bigcup (L_n\cup R_n)$
is a
$\Sigma ^0_2$
-set.
Notation 3.11. For a set
$B\subseteq C(I)\times I$
let us define
$$ \begin{align*} B^{\rightarrow}&=\{(f, f(x))\colon (f, x)\in B\}, \\[-1pt] B^{\leftarrow}&=\{(f, x)\colon (f,f(x))\in B\}, \\[-1pt] B^{\Leftarrow}&=\{(f, x)\colon x\in Acc(f^{-1}(y)), (f,y)\in B\}. \end{align*} $$
Lemma 3.12. Let
$B\subseteq C(I)\times I$
be a Borel set with countable vertical sections. Then the sets
$B^{\rightarrow }, B^{\leftarrow }$
, and
$B^{\Leftarrow }$
are Borel as well.
Proof The evaluation mapping
$e\colon C(I)\times I\to I$
,
$e(f, x)=f(x)$
is continuous. Hence the mapping
$\Phi \colon (f,x)\mapsto (f, e(f,x))$
is continuous as well. Especially, the restriction of
$\Phi $
to B is Borel and also countable-to-1. Since by [Reference Kechris16, Exercise 18.14] countable-to-1 image of a Borel set is Borel we conclude that
$\Phi (B)=B^{\rightarrow }$
is Borel.
By the Lusin–Novikov selection theorem we can write
$B=\bigcup F_n$
for some Borel maps
$F_n$
. It follows then that
$$\begin{align*}B^{\leftarrow}=\bigcup_{n\in\mathbb N}\{(f, x)\colon e(f,x)=F_n(f)\}\\[-15pt] \end{align*}$$
and thus it is a Borel set.
The mapping
$p\colon C(I)\times I\to \mathcal K(I)$
,
$p(f, y)=f^{-1}(y)$
is upper semicontinuous and hence it is Borel by [Reference Kechris16, Exercise 25.14]. The set
$\Gamma $
is Borel by Lemma 3.10 and it has nonempty and countable vertical sections. Hence
$\Gamma =\bigcup b_n$
for some Borel mappings
$b_n\colon \mathcal K(I)\to I$
, by the Lusin–Novikov selection theorem. The mapping
$\Psi _n\colon (f,y)\mapsto (f, b_n(f^{-1}(y)))=(f, b_n(p(f, y)))$
is a Borel mapping and its restriction to B is countable-to-1. Hence by [Reference Kechris16, 18.14] the set
$\bigcup \Psi _n(B)=B^{\Leftarrow }$
is Borel.
Lemma 3.13. Let
$X, Y, Z$
be standard Borel spaces,
$f\colon X\to Y$
a Borel mapping, and
$R{\kern-1.2pt}\subseteq{\kern-1.2pt} Y{\kern-1.2pt}\times{\kern-1.2pt} Z$
a Borel binary relation. Then the set
$R\circ f{\kern-1.2pt}={\kern-1.2pt}\{(x, z)\colon (f(x), z)\in R\}$
is Borel.
Proof Define
$F\colon X\times Z\to Y\times Z$
by
$F(x,z)=(f(x), z)$
. Clearly F is a Borel mapping and
$R\circ f=F^{-1}(R)$
which is consequently a Borel set.
Lemma 3.14. The set
$$ \begin{align*} A=\{(f, x)\in C(I)\times I\colon x\in C_f\cup D_f\} \end{align*} $$
is a Borel subset of
$C(I)\times I$
.
Proof Let us prove first that the set
$B_a:=\{(f,x)\colon x\in M_f\}$
is Borel. As the set
$\{(f, x)\colon x \text { is a left sharp local maximum}\}$
can be written in the form
$$\begin{align*}\bigcup_{\varepsilon>0}\bigcap_{\eta>0}\bigcup_{\delta>0}\{(f,x)\colon \forall z\in [x-\varepsilon, x-\eta]: f(z)\leq f(x)-\delta \And \forall z\in [x,x+\varepsilon]: f(z)\leq f(x)\}, \end{align*}$$
it follows that it is a
$\Sigma ^0_3$
set. By symmetry it follows that
$B_a$
is the union of four
$\Sigma ^0_3$
-sets and thus it is Borel. The set
$B_b:=\{(f, y)\colon f^{-1}(y) \text { contains an interval} \}$
is a
$\Sigma ^0_2$
-set. Let
$B_c:=\{(f, x)\in C(I)\times I\colon x\in \text {Acc}(\text {Fix}(f^n)), n\in \mathbb {N}\}$
. The mapping
$F_n\colon C(I)\to \mathcal K(I)$
,
$F_n(f)=\text {Fix}(f^n)$
is upper semicontinuous (since if
$f_i$
converge uniformly to f and
$x_i$
converge to x with
$f_i^n(x_i)=x_i$
then
$f^n(x)=\lim _i f_i^n(\lim _j x_j)=\lim _j\lim _i f_i^n(x_j)=\lim _j f_j^n(x_j)=\lim _j x_j=x$
by the Moore–Osgood theorem) and thus it is Borel. Since
$\Gamma $
is a Borel set by Lemma 3.10 we conclude that the composition
$\Gamma \circ F_n$
is Borel because the composition of a Borel binary relation and a Borel mapping (in that order) is a Borel relation by Lemma 3.13. Hence
$B_c=\bigcup _{n\in \mathbb N} \Gamma \circ F_n$
is Borel. Hence the set
$B=B_a\cup B_b\cup B_c$
is Borel. Define recursively
$B_1=B$
,
$B_{n+1}=B_n \cup B_n^{\rightarrow } \cup B_n^{\Leftarrow }$
for
$n\in \mathbb N$
. All these sets are Borel by Lemma 3.12. It follows that
$A_1=\bigcup B_n=\{(f, x)\colon x\in C_f\}$
is Borel.
Since
$A_1$
has countable vertical sections and it is Borel we conclude using Proposition 3.9 that
$A_2=\bigcup _{f\in C(I)}(\{f\}\times \text {Cl}(A_{1,f})) $
is Borel as well. Consequently
$A_3=\{(f, x)\colon x\in E_f\}= (C(I)\times \mathbb Q)\setminus A_2$
is Borel. By Lemma 3.12 we conclude that all the sets
$A_{n+1}=A_n\cup A_n^{\rightarrow }\cup A_n^{\leftarrow }$
,
$n\geq 3$
are Borel. Finally
$A=A_1\cup \bigcup _{n\geq 3} A_n$
is a Borel set.
Theorem 3.15. The orientation preserving conjugacy of interval dynamical systems is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation.
Proof By the result of [Reference Hjorth12, Section 4.2] orientation preserving conjugacy of increasing interval homeomorphisms is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation. Hence especially the
$S_{\infty }$
-universal orbit equivalence relation is Borel reducible to increasing conjugacy of orientation preserving interval dynamical systems.
Let us argue for the converse. The set A from Lemma 3.14 is Borel and it has nonempty and countable vertical sections. Hence by the Lusin–Novikov selection theorem we can find Borel mappings
$F_n\colon C(I)\to I$
such that
$\bigcup F_n=A$
. Since all the vertical sections are infinite we can additionally suppose that for every pair
$(f,x)\in A$
there is exactly one
$n\in \mathbb N$
satisfying
$F_n(f)=x$
. Let
$$\begin{align*}\Phi(f)=(\mathbb N, R, m), \end{align*}$$
where R is a binary relation and m is a unary function such that
$aRb$
iff
$F_a(f)\leq F_b(f)$
and
$m(a)=b$
iff
$f(F_a(f))=F_b(f)$
for
$a, b\in \mathbb N$
. There is a natural isomorphism
$\Phi (f)\to \Psi (f)$
,
$a\mapsto F_a(f)$
. Hence clearly
$\Phi $
is a reduction. It is routine to verify that
$\Phi $
is Borel by the fact that the mappings
$F_n$
are Borel.
Let us note that the same conclusion as in the previous theorem can be proved without assuming orientation preserving conjugacy but with just conjugacy. The reason is that in the proofs of Theorem 3.15 and Theorem 3.8 we can simply consider a ternary betweenness relation T instead of the binary relation of linear order
$\leq $
, i.e.,
$(x,y,z)\in T$
if and only if y is an element of the smallest interval containing x and z. This ternary relation is clearly forgetting the order of I. Also by [Reference Hjorth12, Exercise 4.14]
$E_{S_{\infty }}$
is Borel reducible to conjugacy of interval homeomorphisms. Thus we get the following result.
Theorem 3.16. The conjugacy of interval dynamical systems is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation.
We note that Theorem 3.16 is a special case of Hjorth’s conjecture [Reference Hjorth12, Conjecture 10.6] stating that every equivalence relation induced by a continuous action of the group
$\mathcal H(I)$
of all interval homeomorphisms on a Polish space is classifiable by countable structures. In this case the homeomorphism group acts on the space of continuous selfmaps by conjugacy. Similarly one can prove that the orbit equivalence relations induced by natural left or right composition actions of the homeomorphism group on the space of continuous selfmaps are Borel reducible to the
$S_{\infty }$
-universal equivalence relation. Also it is known that the orbit equivalence induced by the homeomorphism group action
$\mathcal H(I)$
on the hyperspace
$\mathcal K(I)$
is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation (see [Reference Hjorth12, Exercise 4.13] or [Reference Chang and Gao5] for a proof). All these are special cases of Hjorth’s conjecture.
4 Hilbert cube dynamical systems
Since the homeomorphism equivalence relation of metrizable compacta is known to be Borel bireducible to the universal orbit equivalence relation, it is not surprising that conjugacy of dynamical systems on the Hilbert cube is of the same complexity, which is the main result of this section. In a dynamical system
$(X, f)$
, a point x is called a locally attracting fixed point if
$f(x)=x$
and there is a neighborhood U of x such that for every
$z\in U$
the trajectory
$(f^n(z))_{n\in \mathbb N}$
converges to x. The notion of a Z-set in the Hilbert cube Q plays an important role and it describes a kind of relative homotopical smallness.
Definition 4.1. A closed subset A of a (separable metric) space X is called a Z-set in X if for every open cover
$\mathcal U$
of X and every continuous function f of the Hilbert cube Q into X there is a continuous function
$g\colon Q\to X$
such that f and g are
$\mathcal U$
-close (i.e., for every
$x\in X$
there is
$U\in \mathcal U$
such that
$f(x), g(x)\in U$
) and
$g(Q)\cap A=\emptyset $
.
An introduction to this notion can be found in [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Chapter 5]. Mostly, we will need the following properties on Z-sets in the Hilbert cube. First, every homeomorphism of Z-sets can be extended to a homeomorphism of the Hilbert cube [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Theorem 5.3.7]. Second, the Hilbert cube
$Q\times I$
contains a topological copy of itself
$Q\times \{0\}$
as a Z-set [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Lemma 5.1.3] and similarly the base in the cone of the Hilbert cube is a Z-set. Third, every closed subset of a Z-set in Q is a Z-set in Q [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Lemma 5.1.2]. If follows from the first and second properties that there is topologically just one way to embed the Hilbert cube into itself as a Z-set (namely
$Q\times \{0\}$
included in
$Q\times I$
). For the purpose of this paper, an absolute retract is just a space homeomorphic to a retract of the Hilbert cube (which is equivalent to being a retract of every separable metric space, in which it is embedded). A space X is said to have the disjoint cell property if for every
$\varepsilon>0$
,
$n\in \mathbb N$
, and continuous mappings
$f, g:I^n\to X$
there are continuous mappings
$f', g': I^n\to X$
with disjoint images such that f and
$f'$
as well as g and
$g'$
are
$\varepsilon $
-close. The last two notions give a topological characterization of the Hilbert cube.
Theorem 4.2 (Toruńczyk; see, e.g., [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Theorem 4.2.25]).
A space X is homeomorphic to the Hilbert cube if and only if it is an absolute retract with the disjoint cell property.
The following proposition is a special case of [Reference Zielinski29, Proposition 1] and it can be easily proved using the back and forth argument. Another reference for the proof is [Reference Lorch20, Propositions 9 and 10]. Our formulation is using a slightly different language.
Proposition 4.3. Let
$K\subseteq A, L\subseteq B$
be four nonempty compact metrizable spaces such that
$A\setminus K$
and
$B\setminus L$
are dense sets of isolated points in A and B respectively. Then every homeomorphism of K onto L can be extended to a homeomorphism of A onto B.
The following will be useful in the proof of Theorem 4.10.
Proposition 4.4 [Reference Gladdines and van Mill10, Theorem 2.6] or [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Corollary 4.2.24].
If X is a nondegenerate Peano continuum then there exists a homotopy
$H\colon \mathcal K(X)\times I\to \mathcal K(X)$
for which:
-
•
$H(A, 0)=A$
for every
$A\in 2^X$
, -
•
$H(A, t)$
is finite for every
$t>0$
and
$A\in 2^X$
.
Recall that if
$Y\subseteq X$
and
$\varepsilon>0$
we say that X is
$\varepsilon $
-deformable into Y if there exists a continuous mapping
$\varphi \colon X\times I\to X$
such that
$\varphi (x,0)=x$
,
$\varphi (x,1)\in Y$
and the diameter of
$\varphi (\{x\}\times I)$
is at most
$\varepsilon $
for every
$x\in X$
. The following proposition was proved in [Reference Krasinkiewicz18, Theorem 1.1 and Corollary 1.3].
Proposition 4.5. Let X be a compact space such that for every
$\varepsilon>0$
there exists an absolute retract
$Y\subseteq X$
for which X is
$\varepsilon $
-deformable into Y. Then X is an absolute retract.
By a result of [Reference Anderson1] the union of two Hilbert cubes, whose intersection is a Z-set in each of the cubes and which is homeomorphic to the Hilbert cube, is the Hilbert cube again. By the result of [Reference Handel11], even a weaker condition is enough to get the same conclusion:
Proposition 4.6. Let X be a space which is the union of two Hilbert cubes
$Q_1$
and
$Q_2$
. Suppose that
$Q_1\cap Q_2$
is a Hilbert cube which is a Z-set in
$Q_1$
. Then X is a Hilbert cube.
It should be noted here that a space which is the union of two Hilbert cubes intersecting in a Hilbert cube may not be a Hilbert cube [Reference Ryll-Nardzewski27].
Lemma 4.7. Let X be a compact metric space which is the union of Hilbert cubes Q,
$Q_1$
,
$Q_2, \dots $
such that
$Q_i\cap Q_j=\emptyset $
and
$Q\cap Q_i$
is a Hilbert cube which is a Z-set in
$Q_i$
for every
$i, j\in \mathbb N$
,
$i\neq j$
. Suppose moreover that the diameter of
$Q_i$
tends to zero. Then X is homeomorphic to the Hilbert cube as well.
Proof Let us denote
$X_i=Q\cup Q_1\cup \dots \cup Q_i$
and observe that it is homeomorphic to the Hilbert cube for every
$i\in \mathbb N$
by an inductive usage of Proposition 4.6. To make the same conclusion for X we use Toruńczyk’s theorem.
There is topologically just one way to embed the Hilbert cube into itself as a Z-set. Hence every pair
$(Q_i, Q_i\cap Q)$
is equivalent to
$(Q\times I, Q\times \{0\})$
because
$Q_i\cap Q$
is a Z-set in
$Q_i$
and it is homeomorphic to the Hilbert cube. Thus simply there is a homotopy
$h_i\colon Q_i\times I\to Q_i$
such that
$h_i(x,t)=x$
for
$t=0$
or
$x\in Q_i\cap Q$
and
$h_i(x,1)\in Q\cap Q_i$
. Let us denote
$$\begin{align*}s_i(x, t)= \begin{cases} x, & x\in X_i, \\ h_j(x, t), &x\in Q_j,\quad j>i. \end{cases} \end{align*}$$
Since the diameter of
$Q_i$
tends to zero, the diameters of
$s_i(\{x\}\times I)$
are sufficiently small for large i. Hence for every
$\varepsilon>0$
it follows that X is
$\varepsilon $
-deformable into
$X_i$
for some
$i\in \mathbb N$
. Moreover, for every
$i\in \mathbb N$
,
$X_i$
is an absolute retract. Hence X is an absolute retract by Proposition 4.5.
Let us argue that X has the disjoint cell property (see [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, p. 294]). Denote
$r_i(x)=s_i(x, 1)$
. Then
$r_i\colon X\to X_i$
is a retraction. Suppose that
$f, g\colon I^n\to X$
are continuous mappings and
$\varepsilon>0$
. Then for sufficiently large
$i\in \mathbb N$
diameters of
$Q_j$
are smaller than
$\varepsilon $
for
$j\geq i$
. Hence
$r_i$
is
$\varepsilon $
-close to identity on X. Since
$X_i$
is homeomorphic to the Hilbert cube it has the disjoint cell property and thus there are continuous mappings
$f', g'\colon I^n\to X_i$
with disjoint images such that
$r_i\circ f$
and
$f'$
as well as
$r_i\circ g$
and
$g'$
are
$\varepsilon $
-close. It follows by the triangle inequality that f and
$f'$
as well as g and
$g'$
are
$2\varepsilon $
-close. Thus X has the disjoint cell property. As mentioned at the beginning of the proof, by Toruńczyk’s theorem it follows that X is homeomorphic to the Hilbert cube.
An equivalence relation E on a Borel subset Y of a Polish space X is said to be countably separated if there is a sequence
$(Z_n)_{n=1}^{\infty }$
of E-invariant Borel subsets of Y such that for all
$x,y \in Y$
, the points x and y are E-equivalent if and only if the sets
$\{ n \in \mathbb {N} \,; \ x \in Z_n \}$
and
$\{ n \in \mathbb {N} \,; \ y \in Z_n \}$
are equal. A transversal for an equivalence relation
$E\subseteq X\times X$
is a set
$T\subseteq X$
whose intersection with every E-equivalence class is a one point set.
The following proposition by Burgess is a special kind of a selection theorem and it will serve as a useful tool to complete the Borel coding argument, which is by no means straightforward.
Proposition 4.8 [Reference Burgess3].
Let G be a Polish group, let X be a Polish space, and let
$\alpha $
be a continuous action of G on X. Denote by E the orbit equivalence relation induced by
$\alpha $
and let Y be an E-invariant Borel subset of X. Let
$E_Y$
be the restriction of E to Y and assume that
$E_Y$
is countably separated. Then there is a Borel transversal for
$E_Y$
.
In the next proposition, we denote by
$\mathcal K_X(Q)$
the collection of subspaces of Q which are homeomorphic to X. It is known for a long time that this is always a Borel set [Reference Motto Ros25].
Proposition 4.9. There is a Borel mapping
$\gamma \colon \mathcal K_Q(Q) \to \text {Inj}(Q, Q)$
such that the image of
$\gamma (R)$
equals to R.
Proof Let G be the homeomorphism group of Q and let us consider the action
$\alpha $
of G on
$\text {Inj}(Q, Q)$
given by
$g\cdot h= h\circ g^{-1}$
. It follows that the corresponding orbit equivalence relation E induced by
$\alpha $
satisfies that
$f E g$
if and only if the images of f and g are equal. Moreover E is countably separated as if we consider a countable base
$\mathcal B$
of Q and
$Z_B=\{f\in \text {Inj}(Q, Q)\colon Im(f)\cap B\neq \emptyset \}$
for
$B\in \mathcal B$
then
$f \mathbin{E} g$
if and only if
$\{B\in \mathcal B\colon f\in Z_B\}=\{B\in \mathcal B\colon g\in Z_B\}$
and also the sets
$Z_B$
are clearly invariant with respect to E. By a straightforward application of Proposition 4.8 we get that there is a Borel transversal T of E. As the mapping
$\chi \colon f\mapsto Im(f)$
,
$\text {Inj}(Q, Q)\to \mathcal K(Q)$
is Borel (even continuous), the graph of
$\chi $
is a Borel subset of
$\text {Inj}(Q, Q)\times \mathcal K(Q)$
. As moreover T is a Borel subset of the domain of
$\chi $
and
$\chi $
is one-to-one on T it follows that the mapping
$\gamma =(\chi |T)^{-1}$
has a Borel graph and thus it is a Borel mapping. Clearly for every
$R\in \mathcal K_Q(Q)$
we get that
$\gamma (R)$
is an embedding of Q into Q whose image equals R.
Some ideas for the proof of the following come from the paper [Reference Krupski and Vejnar19].
Theorem 4.10. The conjugacy of Hilbert cube homeomorphisms (or selfmaps) is Borel bireducible to
$E_{G_{\infty }}$
.
Proof For one direction it is enough to prove that the homeomorphism equivalence relation of metrizable compacta is Borel reducible to conjugacy of Hilbert cube homeomorphisms because the first relation is Borel bireducible to
$E_{G_{\infty }}$
by the main result of [Reference Zielinski29]. To this end let
$$ \begin{align*} Q&=\{x\in\ell_2\colon 0\leq x_n\leq 1/n\}, \\ Q'&=Q\times I\times \{0\}, \\ Q"&=Q\times I\times [-1,1],\\ Q^{\prime -} &=Q\times I\times [-1,0], \end{align*} $$
and let
$\|\cdot \|$
be the usual norm on
$\ell _2$
.
Let us fix a homotopy
$H\colon \mathcal K(Q)\times I\to \mathcal K(Q)$
given by Proposition 4.4 for the case
$X=Q$
. Let us fix
$K\in \mathcal K(Q)$
. We want to find a homeomorphism
$f_K$
of a Hilbert cube
$Q_K\subseteq Q"$
such that the topological information about K is somehow encoded in the dynamics of
$f_K$
. Let
$D_n^K=H(K, 2^{-n})$
,
$n\in \mathbb N$
. Let
$\varepsilon _n$
be the minimum of
$1/n$
and the smallest distance of different points in
$D_n^K$
. For every
$d\in D_n^K$
fix a set
$$\begin{align*}B^d_n=\{(x, 2^{-n},0)\in Q"\colon \|d- x\|\leq\varepsilon_n/3\}. \end{align*}$$
It follows that
$B^d_n$
is always homeomorphic to the Hilbert cube since it is affinely homeomorphic to an infinite dimensional compact convex subset of a Hilbert space [Reference Keller17]. Let
$C^d_n$
be the cone in
$Q"$
with base
$B^d_n$
and with the vertex
$(d, 2^{-n}, 2^{-n})$
,
$d\in D_n^K$
,
$n\in \mathbb N$
, i.e., the union of all segments with end points
$(d, 2^{-n}, 2^{-n})$
and p,
$p\in B^d_n$
. The cone over the Hilbert cube is homeomorphic to the Hilbert cube [Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Theorem 1.7.5], which applies to
$C^d_n$
. Let
$Q_K^m=Q^{\prime -} \cup \bigcup \{ C^d_n\colon n\in \mathbb N, n\leq m, d\in D_n^K\}$
and
$Q_K=\bigcup \{Q_K^m\colon m\in \mathbb N\}$
(see Figure 1). Note that
$Q_K$
is a closed subset of
$Q"$
.
The compactum
$Q_K$
.
Since
$Q^{\prime -} \cap C^d_n=B^d_n$
is homeomorphic to the Hilbert cube, which is a Z-set in
$C^d_n$
, we inductively obtain by Lemma 4.7 that
$Q_K$
is a Hilbert cube.
Let
$h(x)=\sqrt {x}, x\in I$
or any fixed homeomorphism of I with two fixed points
$0,1$
; and
$1$
being a locally attracting fixed point. We define
$$\begin{align*}f_K(x)=\begin{cases}x, &x\in Q^{\prime -}, \\ ((1-h(t))a+h(t)d, 2^{-n}, 2^{-n} h(t)), & \begin{matrix} x=((1-t)a+td, 2^{-n}, 2^{-n}t)\in C^d_n,\\ d\in D_n^K,t\in I, n\in\mathbb{N}. \end{matrix} \end{cases}\end{align*}$$
All the points in
$Q^{\prime -}$
are fixed points for
$f_K$
and these are clearly not attracting. All the points in
$\bigcup D_n^K$
are fixed points of
$f_K$
and these are attracting. There are no other fixed points of
$f_K$
. It follows that K is homeomorphic (or even equal) to the set of fixed points that are limits of attracting points but not attracting by itself (and thus defined only by dynamical notions). Hence if
$f_K$
and
$f_L$
are conjugate then K and L are homeomorphic,
$K, L\in \mathcal K(Q)$
.
On the other hand if
$K, L$
are homeomorphic compacta in Q then the sets
$K\cup \bigcup _{n\in \mathbb N}D_n^K\times \{2^{-n}\}$
and
$L\cup \bigcup _{n\in \mathbb N}D_n^L\times \{2^{-n}\}$
are homeomorphic by Lemma 4.3. This homeomorphism can be simply extended to a homeomorphism
$$\begin{align*}\varphi\colon (K\times \{(0,0)\}) &\cup \bigcup\{B^d_n(K)\colon d\in D_n^K, n\in\mathbb N\} \to (L\times\{(0, 0)\})\\ & \cup \bigcup \{B^d_n(L)\colon d\in D_n^L, n\in\mathbb N\}. \end{align*}$$
Both the sets in the domain and range of
$\varphi $
are Z-sets in
$Q^{\prime -}$
since these are closed subsets of the Z-set
$Q'\times \{0\}$
[Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Lemmas 5.1.2 and Lemma 5.1.3]. Hence
$\varphi $
can be extended to a homeomorphism
$\varphi '\colon Q^{\prime -}\to Q^{\prime -}$
[Reference Thomas, Downey, Brendle, Goldblatt and Kim22, Theorem 5.3.7]. It remains to extend
$\varphi '$
linearly on the cones to obtain a homeomorphism
$\varphi "$
. It follows that
$\varphi "$
conjugates
$f_K$
and
$f_L$
. Note that we can identify
$f_K$
with its graph and thus it can be considered as a closed subspace of
$Q"\times Q"$
. To verify that the mappings
$K\mapsto Q_K$
and
$\chi \colon \mathcal K(Q)\to \mathcal K(Q"\times Q")$
,
$K\mapsto f_K$
are Borel is a routine which is usually omitted in this type of proof.
However, we are still not done, since
$f_K$
is defined on the topological copy of the Hilbert cube
$Q_K$
which differs when changing K. Let us consider the Borel mapping
$\gamma $
given by Proposition 4.9. We redefine the mapping
$f_K$
by conjugating it via
$\gamma (Q_K)$
in the following way. The mapping
$K\mapsto \gamma (Q_K)^{-1} \circ f_K\circ (\gamma (Q_K))$
,
$\mathcal K(Q)\to \mathcal H(Q)$
is the desired Borel reduction.
To conclude the proof it is enough to Borel reduce conjugacy of Hilbert cube maps to
$E_{G_{\infty }}$
. Consider structures of the form
$(Q, R)$
where R is a closed binary relation on Q. Two such structures
$(Q, R)$
and
$(Q, S)$
are said to be isomorphic if there is a homeomorphism
$\psi \colon Q\to Q$
for which
$(\psi \times \psi )(R)=S$
. By a fairly more general result [Reference Rosendal and Zielinski24] it follows that such isomorphism equivalence relation is Borel reducible to
$E_{G_{\infty }}$
. There is a Borel (even continuous) reduction which takes a continuous map
$f\colon Q\to Q$
and assigns
$(Q, \text {graph}(f))$
to it. Combining the two reductions we get the desired one.
5 Concluding remarks and questions
Let us summarize some of the results on the complexity of conjugacy equivalence relation in Table 1 in which we consider conjugacy equivalence relation of maps, homeomorphisms, and pointed transitive homeomorphisms of the interval, circle, Cantor set, and Hilbert cube, respectively. Let us recall that a pointed dynamical system is a triple
$(X, f, x)$
, where
$(X, f)$
is a dynamical system and
$x\in X$
. We say that a pointed dynamical system
$(X, f, x)$
is transitive if the forward orbit of x in
$(X, f)$
is dense. Two pointed dynamical systems
$(X, f, x)$
and
$(Y, g, y)$
are called conjugate if there is a conjugacy of
$(X,f)$
and
$(Y, g)$
mapping x to y. We proceed by a series of simple notes as comments on Table 1.
The complexity of conjugacy equivalence relation.
Note 5.1. Conjugacy of pointed transitive maps of the interval is smooth; indeed it is enough to assign to every pointed transitive dynamical system
$(I, f, x)$
the
$\mathbb {N}\times \mathbb {N}$
matrix of true and false:
$(f^m(x)<f^n(x))_{m, n\in {\mathbb N}}$
which determines f uniquely up to increasing conjugacy.
Note 5.2. There are no transitive homeomorphisms on the interval.
Note 5.3. The complexity result by Hjorth [Reference Hjorth12, Section 4.2] that conjugacy of interval homeomorphisms is Borel bireducible to
$E_{S_{\infty }}$
, remains true for circle homeomorphisms simply by a modification of the original proof. A modification of the proof of Theorem 3.16 will give a similar result for circle maps. The same method as for the interval case can be used just by considering left or right local maxima and minima defined in an obvious way and then iterating this set forward and backward in a similar manner as in Notation 3.3. Thus conjugacy of circle selfmaps is Borel bireducible to the
$S_{\infty }$
-universal orbit equivalence relation.
Note 5.4. Transitive homeomorphisms of the circle are well known to be conjugate to irrational rotations. Hence the rotation number is a complete invariant and hence conjugacy of (pointed) transitive homeomorphisms of the circle is Borel bireducible to the equality on irrationals (or on an uncountable Polish space).
Note 5.5. By a result of Kaya [Reference Kaya14], conjugacy of pointed minimal homeomorphisms of the Cantor set is Borel bireducible to the equality of countable sets
$E_{=^+}$
. Note that his proof works in the same vein for pointed transitive homeomorphisms of the Cantor set. Let us recall the main part of his construction in this case. Let X be the Cantor set and
$\mathcal B$
the collection of all clopen sets in X. To a pointed transitive system
$(X, f, x)$
we assign the collection
$$\begin{align*}\mathsf{Ret}(f, x)=\{\mathsf{Ret_B}(f, x)\colon B\in\mathcal B\}, \end{align*}$$
where
$\mathsf {Ret_B}(f, x)=\{n\in \mathbb {Z}\colon f^n(x)\in B\}$
. It can be verified that the mapping
$\Phi $
defined as
$$\begin{align*}\Phi(f, x)=(\mathsf{Ret_B}(f, x)\colon B\in\mathcal B)\in \mathcal P(\mathbb{Z})^{\mathcal B} \end{align*}$$
is a reduction of pointed transitive Cantor maps to the equality of countable sets in
$\mathcal P(\mathbb {Z})^{\mathcal B}$
, i.e.,
$(f,x)$
is conjugate to
$(g, y)$
if and only if
$\mathsf {Ret}(f,x)=\mathsf {Ret}(g,y)$
.
The following question is the missing part to complete Table 1.
Question 5.6. What is the complexity of conjugacy of transitive pointed Hilbert cube homeomorphisms (or maps)?
It was explained to us by Kaya [Reference Kaya15] that conjugacy equivalence relation of pointed transitive Hilbert cube homeomorphisms is a Borel relation. The main reason is that every conjugacy of such systems preserves the distinguished point and thus it is automatically prescribed on a dense subset. Hence there is at most one conjugacy between such systems. Let us note that neither
$E_{S_{\infty }}$
nor
$E_{G_{\infty }}$
is Borel and thus these equivalence relations cannot answer Question 5.6.
Since triangular maps, i.e., maps
$f\colon I^2\to I^2$
of the form
$f(x,y)=(g(x), h(x,y))$
for continuous maps
$g\colon I\to I$
and
$h\colon I^2\to I$
, lie in between one-dimensional and two-dimensional and there is a gap in the complexity of the last two mentioned equivalence relations, the following question is natural.
Question 5.7. What is the complexity of conjugacy of triangular maps? Is it Borel bireducible to
$E_{S_{\infty }}$
or to
$E_{G_{\infty }}$
?
Positive answer to the next question would provide a strengthening of Theorem 3.16.
Question 5.8. Is conjugacy of closed binary relations on the closed interval Borel reducible to the
$S_{\infty }$
-universal orbit equivalence relation?
The answer to the preceding question is affirmative if Hjorth’s conjecture is true.
Funding
This work has been supported by Charles University Research Centre program No.UNCE/SCI/022.
