1 Introduction
Let
$X=\{X_{n}\}$
be a Markov chain moving through a measurable space E and consider a family
$\{f_{\alpha }\}_{\alpha \in E}$
of homeomorphisms of the circle
$S^{1}$
. Under appropriate measurability assumptions, these two ingredients specify, for every x, a Markovian random iteration given by
$$ \begin{align*}f_{X}^{n}(x)= f_{X_{n-1}}\circ \cdots\circ f_{X_{0}}(x). \end{align*} $$
For an independent and identically distributed (i.i.d.) sequence
$\{X_{n}\}$
, it was proved by Malicet in [Reference Malicet24] that if the maps
$f_{\alpha }$
do not have an invariant measure in common, then there is
$\rho <1$
such that, for every
$x\in S^{1}$
, with probability 1, there is an interval I containing x such that
$$ \begin{align} \mbox{diam}\, f_{X}^{n}(I)\leq \rho^{n} \quad \mbox{for every } n\geq 0. \end{align} $$
From this local stability, Malicet deduced several dynamical properties for i.i.d. random iterations of homeomorphisms of the circle. One remarkable property is the generalization of the Antonov’s result [Reference Antonov2, Reference Kleptsyn and Nal’skii21] on global synchronization of random orbits. Also, the local synchronization property (1.1) has found an application in [Reference Szarek and Zdunik30], where the authors obtain a central limit theorem for a certain class of i.i.d. random iterations of homeomorphisms of the circle.
The main result of this paper, Theorem 2.4, extends the local stability (1.1) for a certain class of Markovian random iterations of homeomorphisms of
$S^{1}$
; see §2.3 for the precise statement. It is worth mentioning that this result is new even for Markovian random iterations of
$C^{1}$
-diffeomorphisms of the circle.
There are two classical approaches to the study of Markovian random dynamical systems: a description via the Markov semigroup and a description via a dynamical system called skew product flow; see [Reference Barnsley, Elton and Hardin7, Reference Crauel11, Reference Díaz and Matias13, Reference Edalat14, Reference Kleptsyn and Volk20]. The proof of our main theorem is based on results from these two points of view, which can be linked via a correspondence between them, in a sense that we briefly describe now, in a general setting, for the discrete-time case.
Let
$\{f_{\alpha }\}_{\alpha \in E}$
denote a family of measurable transformations of a standard measurable space M. Assume that the Markov chain
$\{X_{n}\}$
is taken to be in its canonical form, that is,
$X_{n}$
is the natural projection on the product space
$\Sigma =E^{\mathbb {N}}$
(endowed with a Markov measure
$\mathbb {P}$
), given by
$X_{n}(\omega )=\omega _{n}$
. The map
$F\colon \Sigma \times M\to \Sigma \times M$
defined by
$$ \begin{align*}F(\omega,x)=(\sigma(\omega),f_{\omega_{0}}(x)), \end{align*} $$
where
$\sigma $
is the shift map on
$\Sigma $
, is called the skew product associated with the Markovian random iteration
$f_{X}^{n}$
. We also assume that the probability measure
$\mathbb {P}$
is invariant by the shift map on
$\Sigma $
. The skew product approach focuses on the study of the invariant measures for F whose first marginal is
$\mathbb {P}$
. These measures are called invariant measures for the Markovian random iteration.
On the other hand, the Markov semigroup approach proceeds by specifying a transition probability related to the behavior of the Markovian random iteration. In the i.i.d. case, the sequence of random variables
$\{f_{X}^{n}(x)\}$
is a homogeneous Markov chain with transition probability given by
$p(x, A)=\mathbb {P}(f_{X_{0}}(x)\in A)$
. However, for a non-independent Markovian random iteration, the sequence
$\{f_{X}^{n}(x)\}$
is no longer a homogeneous Markov chain. One way to overcome this situation is to consider the joint sequence
$$ \begin{align*}Z_{n}=(X_{n-1}, f_{X}^{n}(x)), \end{align*} $$
which is a homogeneous Markov chain with state space
$E\times M$
with a well-defined transition probability
$\hat p$
; see §2.1. In this setting, the Markov semigroup approach for Markovian random iterations focuses on the study of stationary measures for the transition probability
$\hat p$
.
To sum up, there are two classes of ‘invariant measures’ that we can consider: the stationary measures for
$\hat p$
and the invariant measures for the Markovian random iteration. We observe that in the i.i.d. case there is a simple relation between these two classes. More precisely, a probability measure
$\nu $
on M is a stationary (respectively, ergodic) measure if and only if the product measure
$\mathbb {P}\times \nu $
is an invariant (respectively, ergodic) measure for the skew product; see Ohno [Reference Ohno26]. This result was first proved by Kakutani [Reference Kakutani and Neyman18], under the assumption that the maps have a common fixed point. This correspondence has found applications in several works; see, for instance, [Reference Alves and Araújo1, Reference Araújo3, Reference Araújo and Aytaç4, Reference Bocker-Neto and Viana8, Reference Ledrappier22, Reference Malicet24, Reference Marie and Rousseau25]. See [Reference Kifer19, Reference Viana31] for different proofs of this one-to-one correspondence in the i.i.d. case.
In Theorem 2.1 we establish a one-to-one correspondence between the set of stationary measures for
$\hat p$
and a certain class of invariant measures for the Markovian random iteration, and we also give a useful characterization of stationary measures in terms of the disintegration of measures. We observe that in [Reference Crauel11], Crauel obtained a similar correspondence for the transition probability
$\bar p$
associated to the Markov chain
$\bar {Z}_{n}=(X_{n}, f_{X}^{n}(x))$
. Although this correspondence has a simpler description than the one we obtain in Theorem 2.1 (see Remark 2.2), in this paper we deal with
$\hat p$
because it is a Feller transition probability under weaker assumptions than
$\bar p$
is, which is a regularity needed to prove Theorem 2.4.
It is worth mentioning that in [Reference Hairer16], Hairer introduced the notion of stochastic dynamical system, which is very close to the definition of Markovian random dynamical system introduced in [Reference Crauel11]. Then, for suitable definitions of invariant measure and stationary measure, it is proved that the set of invariant measures and the set of stationary measures are the same, under a certain equivalence relation. This ‘correspondence’ is used to study the ergodic properties of finite-dimensional systems of stochastic differential equations driven by non-degenerate additive fractional Brownian motion; see also [Reference Hairer and Ohashi17].
In this paper we use the correspondence established in Theorem 2.1 to obtain the local exponential synchronization for Markovian random iterations of homeomorphisms of
$S^{1}$
(Theorem 2.4). Roughly speaking, to prove Theorem 2.4 the idea is to reformulate (1.1) in terms of a random ‘Lyapunov exponent’
$\omega \mapsto \lambda (\omega ,x)$
, introduced in [Reference Malicet24], and then the proof is reduced to showing that
$\lambda (\omega ,x)$
is almost surely negative. This is obtained combining via Theorem 2.1 two main results. The first one is a Krylov–Bogolyubov-type result for Markov chains stated in Theorem 4.1, whose proof is a direct consequence of Furstenberg [Reference Furstenberg15, Lemma 2.5]. The second result is the invariance principle for skew products associated with stationary random iterations of homeomorphisms of the circle [Reference Malicet24, Theorem F], which is a modification of Ávila–Viana invariance principle [Reference Ávila and Viana6] (see also Crauel [Reference Crauel10] and Ledrappier [Reference Ledrappier22]).
2 Statement of results
2.1 Preliminary definitions and notation
In this section we state some of the main notions needed throughout this paper.
Let
$(Y,\mathscr {Y})$
be a measurable space and consider a transition probability
$K\colon Y\times \mathscr {Y}\to [0,1]$
, that is, for every
$y\in Y$
, the mapping
$A\mapsto K(y,A)$
is a probability measure on E and, for every
$A\in \mathscr {Y}$
, the mapping
$y \mapsto K(y,A)$
is measurable with respect to
$\mathscr {Y}$
. Recall that a probability measure m on Y is called a stationary measure for the transition probability K if
$$ \begin{align*}m(A)=\int\! K(y,A)\, dm(y) \quad \mbox{for every } A\in \mathscr{Y}. \end{align*} $$
Assume that Y is a compact metric space endowed with its Borel
$\sigma $
-algebra. The transition probability K acts on bounded measurable functions through the mapping
$$ \begin{align*}Kh(y)=\int\! h(x)\, K(y,dx),\quad y\in Y. \end{align*} $$
It is well known that a probability measure
$\nu $
is a stationary measure for K if and only if
$$ \begin{align*}\int\! Kh\, d\nu=\int\! h\,d\nu \end{align*} $$
for every continuous function
$h\colon Y\to \mathbb {R}$
. Also, recall that the transition probability K is said to be:
-
(i) Feller if
$Kh$
is continuous for every continuous function
$h\colon Y\to \mathbb {R}$
; -
(ii) strong Feller if
$Kh$
is continuous for every bounded measurable function
$h\colon Y\to \mathbb {R}$
.
Let
$(E,\mathscr {E})$
be a standard measurable space and consider a transition probability
$p\colon E\times \mathscr {E}\to [0,1]$
. Let
$\Sigma $
be the product space
$E^{\mathbb {N}}$
endowed with the product
$\sigma $
-algebra
$\mathscr {F}=\mathscr {E}^{\mathbb {N}}$
and m a probability measure on E. Associated with the pair
$(p,m)$
there is a unique probability measure
$\mathbb {P}$
on
$\Sigma $
, called Markov measure, for which the sequence of natural projections
$X_{n}(\omega )=\omega _{n}$
is a homogeneous Markov chain on the probability space
$(\Sigma , \mathscr {F}, \mathbb {P})$
with transition probability p and starting measure m; see [Reference Revuz27]. Throughout this paper, we assume that the transition probability p has a unique stationary measure m, and
$\mathbb {P}$
stands for the Markov measure associated with
$(p,m)$
.
Let
$(M, \mathscr {B})$
be a standard measurable space and
$f\colon E\times M\to M$
a measurable map. Denote by
$f_{\alpha }$
the map
$f_{\alpha }(x)=f(\alpha ,x)$
. Then the map
$\varphi \colon \mathbb {N}\times \Sigma \times M\to M$
defined by
$$ \begin{align*}\varphi(n,\omega,x)\stackrel{\scriptscriptstyle\rm def}{=} f_{\omega_{n-1}}\circ\cdots \circ f_{\omega_{0}}(x)\stackrel{\scriptscriptstyle\rm def}{=} f_{\omega}^{n}(x) \end{align*} $$
is called a Markovian random iteration of maps. The term ‘Markovian’ means that in the study of this map under a probabilistic point of view, we consider the space
$\Sigma $
endowed with a Markov measure. We often refer to
$\varphi $
as a Markovian random iteration of maps associated with
$(p,m)$
and f. If M is a topological space, we say that
$\varphi $
is a Markovian random iteration of homeomorphisms of M if the map
$x\mapsto f_{\alpha }(x)$
is a homeomorphism for every
$\alpha $
.
For every
$x\in M$
, the sequence of random variables
$Z_{n}(\omega )=(\omega _{n-1}, f_{\omega }^{n}(x))$
is a homogeneous Markov chain with range on the space
$E\times M$
and transition probability given by
where
indicates the characteristic function of B. A stationary measure for this transition probability is called a stationary measure for
$\varphi $
and the set of stationary measures for
$\varphi $
is denoted by
$S(\varphi )$
.
The skew product associated with
$\varphi $
is the map
$F\colon \Sigma \times M\to \Sigma \times M$
defined by
$$ \begin{align*}F(\omega,x)\stackrel{\scriptscriptstyle\rm def}{=}(\sigma(\omega),f_{\omega_{0}}(x)), \end{align*} $$
where
$\sigma $
is the shift map on
$\Sigma $
. Usually, we simply say that F is the skew product associated with the map f. Recall that a probability measure
$\hat \mu $
is F-invariant if
$F_{*}\hat \mu =\hat \mu $
, where
$*$
denotes the push-forward of a measure.
2.2
$\varphi $
-invariant measures and stationary measures
Let
$(Y, \mathscr {Y})$
be a measurable space and consider a probability measure
$\eta $
on the product space
$Y\times M$
. Let
$\Pi _{1}\colon Y\times M\to Y$
be the natural projection on the first factor given by
$\Pi _{1}(y,x)=y$
. Then there is a unique (almost everywhere) family of probability measures
$\{\eta _{y}\colon y\in Y\}$
on the space M such that, for every measurable rectangle
$A\times B$
, the map
$y\mapsto \eta _{y}(B)$
is measurable and
$$ \begin{align} \eta(A\times B)=\int_{A} \eta_{y}(B)\, d\,\Pi_{1*}\eta. \end{align} $$
The family
$\{\eta _{y}\colon y\in Y\}$
is called the disintegration of
$\eta $
with respect to
$\Pi _{1*}\eta $
; see Rokhlin [Reference Rokhlin28]. The probability measure
$(\Pi _{1})_{*}\eta $
on Y is called the first marginal of
$\eta $
. Conversely, let
$\{\eta _{y}\colon y\in Y\}$
be a family of probability measures on the space M such that, for every measurable set
$B\subset M$
, the map
$y\mapsto \eta _{y}(B)$
is measurable. Then for every probability measure
$\tau $
on Y there is a unique probability measure
$\eta $
on the product space
$Y\times M$
with first marginal
$\tau $
whose disintegration with respect to
$\tau $
is given by this family. Indeed, we define
$\eta $
as in (2.2) with
$\tau $
in place of
$\Pi _{1*}\eta $
.
Let F be the skew product associated with
$\varphi $
. An F-invariant measure with first marginal
$\mathbb {P}$
is called a
$\varphi $
-invariant measure. Let
$I_{0}(\varphi )$
be the set of
$\varphi $
-invariant measures
$\hat \mu $
for which there is a family
$\{\mu _{\alpha }\colon \alpha \in E\}$
of probability measures on M such that
$$ \begin{align*}\hat \mu_{\omega}=\mu_{\omega_{0}} \end{align*} $$
for every
$\omega $
, where the family
$\{\hat \mu _{\omega }\colon \omega \in \Sigma \}$
is the disintegration of
$\hat \mu $
with respect to
$\mathbb {P}$
. In other words,
$I_{0}(\varphi )$
consists of
$\varphi $
-invariant measures whose disintegration depends only on the zeroth coordinate of
$\omega $
.
For the next result, we need the following definition. We say that a transition probability q on E is in duality with the transition probability p relative to the stationary measure m if, for all measurable sets A and B, we have
$$ \begin{align*}\int_{B} p(\alpha,A)\,dm(\alpha)=\int_{A} q(\beta,B)\,dm(\beta). \end{align*} $$
On standard measurable spaces, there is an essentially unique transition probability in duality with p relative to m; see [Reference Revuz27, Lemma 4.7 and Theorem
$4.9$
].
Theorem 2.1. Let
$\varphi $
be a Markovian random iteration associated with
$(p,m)$
and f.
-
(i) A probability measure
$\nu $
on
$E\times M$
is a stationary measure for
$\varphi $
if and only if
$\nu $
has first marginal m and for m-almost every
$$ \begin{align*}\nu_{\alpha}= \int\!f_{\alpha*}\nu_{\beta}\, q(\alpha,d\beta) \end{align*} $$
$\alpha $
, where
$\{\nu _{\alpha }\colon \alpha \in E\}$
is the disintegration of
$\nu $
with respect to m.
-
(ii) There is a one-to-one correspondence between the sets
$S(\varphi )$
and
$I_{0}(\varphi )$
given by and
$$ \begin{align*}\hat \mu\in I_{0}(\varphi)\mapsto \nu\in S(\varphi),\quad \nu_{\alpha}= f_{\alpha*}\mu_{\alpha}, \end{align*} $$
Moreover, this relation defines a one-to-one correspondence between the ergodic stationary measures and the ergodic
$$ \begin{align*}\nu \in S(\varphi)\mapsto \hat \mu \in I_{0}(\varphi),\quad \mu_{\alpha}=\int\!\nu_{\beta}\,q(\alpha,d\beta). \end{align*} $$
$\varphi $
-invariant
Remark 2.2. In the study of Markovian random iterations, alternatively, we can consider the Markov chain
$$ \begin{align*}\bar Z_{n}=(\omega_{n},f_{\omega}^{n}(x)) \end{align*} $$
rather than
$Z_{n}=(\omega _{n-1},f_{\omega }^{n}(x))$
; see, for instance, [Reference Crauel12, Reference Kleptsyn and Volk20]. In this case, the transition probability is given by
We observe that a similar result to Theorem 2.1 could also be obtained for
$\bar p$
, as follows.
-
(i) A probability measure
$\nu $
on
$E\times M$
is a stationary measure for
$\bar p$
if and only if
$\nu $
has first marginal m and for m-almost every
$$ \begin{align*}\nu_{\alpha}=\int\! f_{\beta*}\nu_{\beta}\, q(\alpha,d\beta) \end{align*} $$
$\alpha $
.
-
(ii) There is a one-to-one correspondence between the set of stationary measures for
$\bar p$
and
$I_{0}(\varphi )$
given by and
$$ \begin{align*}\hat \mu\in I_{0}(\varphi)\mapsto \nu\in S(\varphi),\quad \nu_{\alpha}=\mu_{\alpha}, \end{align*} $$
$$ \begin{align*}\nu \in S(\varphi)\mapsto \hat \mu \in I_{0}(\varphi),\quad \mu_{\alpha}=\nu_{\alpha}. \end{align*} $$
The proof of the first item is similar to the proof of item (i) of Theorem 2.1. The second item was already proved by Crauel in [Reference Crauel11] for Markovian random dynamical systems. Nevertheless, an alternative short proof could be obtained using a similar argument to that used to prove item (ii) of Theorem 2.1.
As mentioned in the Introduction, the reason to consider
$\hat p$
instead of
$\bar p$
in this paper is that
$\hat p$
is a Feller transition probability under weaker assumptions than
$\bar p$
is (see Remark 2.5).
Remark 2.3. In [Reference Malheiro and Viana23], Malheiro and Viana consider Markovian random products of finitely many matrices
$A_{1},\ldots , A_{k}$
of
$\mbox {GL}(k)$
and introduce the notion of unit stationary vectors. We observe that item (i) of Remark 2.2 displays a relation between the unit stationary vectors and the stationary measures for
$\bar p$
.
Indeed, let
$P=(p_{ij})$
be a
$k\times k$
irreducible transition matrix and
$p=(p_{1},\ldots , p_{k})$
the unique stationary probability vector. Then a unit stationary vector is defined as a k-tuple
$(\mu _{1},\ldots , \mu _{k})$
of probability measures on
$\mathbb {P}(\mathbb {R}^{k})$
such that
$$ \begin{align*}p_{i}\mu_{i}=\sum_{j=1}^{k} p_{j}p_{ji}A_{j*}\mu_{j}. \end{align*} $$
It turns out that the transition matrix
$Q=(q_{ij})$
in duality with P relative to p is given by
$$ \begin{align*}q_{ij}=\frac{p_{j}}{p_{i}}p_{ji}. \end{align*} $$
Hence, item (i) of Remark 2.2 implies that if
$\nu $
is a probability measure on
$\{1,\ldots , k\}\times \mathbb {P}(\mathbb {R}^{k})$
with first marginal given by
$(p_{1},\ldots , p_{k})$
and disintegration given by the k-tuple
$(\mu _{1},\ldots , \mu _{k})$
, then
$\nu $
is a stationary measure for the transition probability
$\bar p$
(as defined in Remark 2.2) associated with the Markovian random product of matrices given by
$(P,p)$
and
$A_{1},\ldots , A_{k}$
.
2.3 Local synchronization
In this section we state the main result of this paper. To this end, we need to consider a certain boundedness condition on the transition probability p. To introduce this condition, denote by
$p_{\alpha }$
the probability measure
$p_{\alpha }(A)=p(\alpha ,A)$
. We say that the pair
$(p,m)$
is bounded if
$p_{\alpha }$
is absolutely continuous with respect to m for every
$\alpha $
and there is a constant C with
$0<C\leq 1$
such that
$$ \begin{align} C\leq \frac{dp_{\alpha}}{dm}(\beta)\leq C^{-1} \end{align} $$
for all
$\alpha $
and
$\beta $
in E.
In what follows in this section, E is a compact metric space endowed with its Borel
$\sigma $
-algebra.
Theorem 2.4. Consider a Markovian random iteration of homeomorphisms of
$S^{1}$
associated with a bounded pair
$(p,m)$
and
$f\colon E\times S^{1}\to S^{1}$
such that
$\hat p$
is Feller. Assume that there is no probability measure
$\mu $
such that
$f_{\alpha *}\mu =\mu $
for m-almost every
$\alpha $
. Then there is
$\rho <1$
such that, for any
$x\in S^{1}$
, for
$\mathbb {P}$
-almost every
$\omega \in \Sigma $
, there is a neighborhood I of x such that
$$ \begin{align*}\mathrm{diam}\, f_{\omega}^{n}(I)\leq \rho^{n} \quad \mbox{for every } n\geq 0. \end{align*} $$
This synchronization phenomenon for Markovian random iterations of finitely many maps on compact subsets of a finite-dimensional Euclidean space was already studied in [Reference Díaz and Matias13]. Assuming a ‘topological’ condition on the maps, called a splitting condition, the authors proved a contraction, exponentially fast, of the whole space under the action of the Markovian random iteration. However, in the circle
$S^{1}$
, the splitting condition is never satisfied.
Remark 2.5. Let us observe that the transition probability
$\hat p$
(in Theorem 2.4) will be Feller if we assume one of following conditions:
-
(i) p is strong Feller;
-
(ii) p is Feller and for every
$x\in S^{1}$
, the map
$\alpha \mapsto f_{\alpha }(x)$
is continuous.
Indeed, let
$\hat \phi \colon E\times S^{1}\to \mathbb {R}$
be a continuous function. If
$(\alpha _{n},x_{n})\to (\alpha ,x)$
, we have
$$ \begin{align*} &|\hat p\hat \phi(\alpha_{n},x_{n})-\hat p\hat \phi(\alpha,x)|\\ &\quad =\bigg|\int\! \hat \phi(\beta, f_{\beta}(x_{n}))\,p(\alpha_{n},d\beta)-\int\! \hat \phi(\beta, f_{\beta}(x))\,p(\alpha,d\beta)\bigg|\\ &\quad \leq \int\! |\hat \phi(\beta, f_{\beta}(x_{n}))-\hat \phi(\beta, f_{\beta}(x))|\,p(\alpha_{n},d\beta)+|p\phi_{x}(\alpha_{n})-p\phi_{x}(\alpha)|\\ &\quad \leq C^{-1} \int\! |\hat \phi(\beta, f_{\beta}(x_{n}))-\hat \phi(\beta, f_{\beta}(x))|\,dm(\beta)+|p\phi_{x}(\alpha_{n})-p\phi_{x}(\alpha)|, \end{align*} $$
where
$\phi _{x}(\beta )=\hat \phi (\beta ,f_{\beta }(x))$
and we used that
$(p,m)$
is bounded. Now, any of the above conditions implies that
$p\phi _{x}$
is continuous. Hence, using that the map
$x\mapsto f_{\beta }(x)$
is continuous for every
$\beta $
, we can apply the dominated convergence theorem to get that
$\hat p\hat \phi $
is continuous.
Note that none of the above conditions guarantees that the transition probability
$\bar p$
(as defined in Remark 2.2) is Feller. Indeed, let
$g\colon S^{1}\to \mathbb {R}$
be a continuous function and define
$\hat g\colon E\times S^{1}\to \mathbb {R}$
by
$\hat g(\alpha ,x)=g(x)$
. Then
$$ \begin{align*}\bar p\hat g(\alpha,x)=\int\! \hat g(\beta, f_{\alpha}(x))\, p(\alpha,d\beta)=g(f_{\alpha}(x)). \end{align*} $$
In particular,
$\bar p \hat g$
may be discontinuous if f is not a jointly continuous function.
We now present some examples of bounded pairs
$(p,m)$
such that p is strong Feller.
Example 2.1 (Random iteration driven by random walks). Consider a sequence
$\{W_{n}\}_{n\geq 0}$
of i.i.d. random variables taking values in
$S^{1}=\mathbb {R}/\mathbb {Z}$
whose common distribution is the Lebesgue measure
$\mathrm {Leb}$
of
$S^{1}$
. Then the sequence
$$ \begin{align*}X_{n}=W_{1}+\cdots+W_{n} \end{align*} $$
is a Markov chain with transition probability given by
$p(x, A)=\mathrm {Leb}(A)$
. In particular, p is strong Feller. Note that
$\mathrm {Leb}$
is a stationary measure for p. The Radon–Nikodym derivative of
$p_{x}$
with respect to the Lebesgue measure is equal to
$1$
. Hence, the pair
$(p,\mathrm {Leb})$
is bounded.
More generally, let G be a compact topological group endowed with its Borel
$\sigma $
-algebra
$\mathscr {G}$
. Recall that a left random walk on G with law
$\mu $
is a Markov chain
$\{X_{n}\}$
with transition probability given by
$$ \begin{align*}p(g,A)=\mu\ast\delta_{g}(A) \quad \mbox{for every } A\in \mathscr{G}, \end{align*} $$
where
$\ast $
denotes the convolution of measures. We observe that the Haar measure m of G is a stationary measure for p. Let us assume that
$\mu $
is absolutely continuous with respect to m and let f denote the Radon–Nikodym derivative
${d\mu }/{dm}$
. We claim that
$$ \begin{align*}\frac{dp_{g}}{dm}(h)=f(hg^{-1}). \end{align*} $$
Indeed, we have
where in the fourth equality we use that the Haar measure on a compact topological group is invariant for left and right translations.
In particular, if there is
$C>0$
such that the Radon–Nikodym derivative f satisfies
$$ \begin{align*}C\leq f(g)\leq C^{-1} \end{align*} $$
for every
$g\in G$
, then the pair
$(p,m)$
is bounded. Moreover, if f is continuous we have that p is strong Feller.
Example 2.2 (Markovian random iterations of finitely many maps). Consider a transition matrix
$P=(p_{ij})$
with positive entries, which is clearly strong Feller. It follows from the Perron–Frobenius theorem that the matrix P has a unique stationary probability vector
$p=(p_{i})$
with positive entries. In particular, the pair
$(P,p)$
is bounded.
3 Proof of Theorem 2.1
In this section
$(M,\mathscr {B})$
and
$(E,\mathscr {E})$
are standard measurable spaces,
$f\colon E\times M\to M$
is a measurable map, p is a transition probability on E with a unique stationary measure m and
$\varphi $
is the Markovian random iteration associated with
$(p,m)$
and f.
3.1 Proof of Theorem 2.1
We first prove item (i). We start by showing that any stationary measure for
$\varphi $
has first marginal m.
Let
$\hat p$
be the transition probability on
$E\times M$
induced by
$\varphi $
as defined in (2.1) and denote by
$m'$
the first marginal of
$\nu $
. Then, given a measurable set
$A\in \mathscr {E}$
, we have
Hence,
$m'$
is a stationary measure for p. Since we are assuming that m is the unique stationary measure for p, we conclude that
$m'=m$
.
Now, let
$\nu $
be a probability measure on
$E\times M$
with first marginal m. We have
Next, we need the following well-known property of transition probabilities in duality: for every measurable map
$\kappa \colon E\times E\to [0,\infty )$
, we have
$$ \begin{align} \int\! \kappa(\alpha,\beta)\, p(\alpha,d\beta)\, dm(\alpha)=\int\! \kappa(\beta,\alpha)\, q(\alpha,d\beta)\, dm(\alpha). \end{align} $$
Applying this property, we obtain
This implies that
$\nu $
is stationary if and only if
$\nu $
has first marginal m and
$$ \begin{align*}\nu_{\alpha}=\int\! f_{\alpha*}\nu_{\beta}\, q(\alpha,d\beta) \end{align*} $$
for m-almost every
$\alpha $
. This completes the proof of item (i).
Now we prove item (ii). First, let
$\Theta $
be the map that associates to each
$\hat \mu \in I_{0}(\varphi )$
the probability measure on
$E\times M$
given by
$$ \begin{align*}\Theta(\hat\mu)=\nu \quad \mbox{where } \nu_{\alpha}=f_{\alpha*}\mu_{\alpha}. \end{align*} $$
We need to show that
$\Theta $
takes
$I_{0}(\varphi )$
into
$S(\varphi )$
and that if
$\mu $
is an ergodic F-invariant measure, then
$\nu $
is an ergodic stationary measure for
$\hat p$
. To this end, for
$B\in \mathscr {E}\otimes \mathscr {B}$
, define
$\hat B\in \Sigma \times M$
by
$\hat B\stackrel {\scriptscriptstyle \rm def}{=}\{(\omega ,x)\colon (\omega _{0},f_{\omega _{0}}(x))\in B\}$
. Note that
$\hat B\in \mathscr {F}\otimes \mathscr {B}$
and the section
$\hat B^{\omega }$
coincides with
$f_{\omega _{0}}^{-1}(B^{\omega _{0}})$
for every
$\omega $
. Thus, for
$\hat \mu \in I_{0}(\varphi )$
, we have
$$ \begin{align*}\hat\mu(B)=\!\!\int\! \hat \mu_{\omega}(\hat{B}^{\omega})\, d\mathbb{P}(\omega)=\!\!\int\! \mu_{\omega_{0}}(f_{\omega_{0}}^{-1}(B^{\omega_{0}}))\, d\mathbb{P}(\omega) =\int\! f_{\omega_{0*}}\mu_{\omega_{0}}(B^{\omega_{0}})\, dm(\omega_{0})=\nu(B), \end{align*} $$
where
$\nu =\Theta (\hat \mu )$
. Therefore,
which implies that
$\nu $
is a stationary measure for
$\hat p$
. Now assume that
$\hat \mu $
is an ergodic F-invariant measure. Let
$B\subset E\times M$
be a measurable set such that
for
$\nu $
-almost every
$(\alpha ,x)$
. We claim that
$\nu (B)\in \{0,1\}$
, which proves that
$\nu $
is ergodic. To prove this claim, note that by the definition of
$\hat p$
and the assumption on the set B, we have that, for
$\nu $
-almost every
$(\alpha ,x)$
,
Therefore,
which implies that
for
$\hat \mu $
-almost every
$(\omega ,x)$
. Since
$\hat \mu $
is ergodic, we have
$\nu (B)=\hat \mu (\hat B)\in \{0,1\}$
.
Now let
$\Xi $
be the map that associates to each probability measure
$\nu $
on
$E\times M$
with first marginal m the probability measure on
$\Sigma \times M$
given by
$$ \begin{align*}\Xi(\nu)=\hat \mu \quad\mbox{where } \hat \mu_{\omega}=\nu_{\omega_{0}}. \end{align*} $$
We need to show that
$\Xi $
takes
$S(\varphi )$
into
$I_{0}(\varphi )$
and that if
$\nu $
is an ergodic stationary measure for
$\hat p$
, then
$\hat \mu =\Xi (\nu )$
is an ergodic F-invariant measure.
Since
$(E, \mathscr {E})$
is assumed to be a standard measurable space, it is well known (see Kifer [Reference Kifer19]) that there exist a probability space
$(R,\mathscr {R}, \rho )$
and a measurable function
$g\colon R\times E \to E$
such that
Consider the map
$h\colon R\times E\times M\to E\times M$
given by
$$ \begin{align*}h(r, \alpha,x)=( g(r,\alpha),f_{g(r,\alpha)}(x)). \end{align*} $$
Then the i.i.d. random iteration of maps
$\hat \varphi \colon \mathbb {N}\times R^{\mathbb {N}}\times E\times M\to E\times M $
associated with h and
$(R,\mathscr {R}, \rho )$
has transition probability given by
$\hat p$
. Let
$G\colon R^{\mathbb {N}}\times E\times M\to R^{\mathbb {N}}\times E\times M$
be the skew product associated with
$\hat \varphi $
.
Lemma 3.1. There is a measurable map
$\pi \colon R^{\mathbb {N}}\times E\times M\to \Sigma \times M$
such that
$$ \begin{align*}\pi\circ G=F\circ \pi \end{align*} $$
and, for every probability measure
$\nu $
on
$E\times M$
with first marginal m, we have
$\pi _{*}(\rho ^{\mathbb {N}}\times \nu )=\Xi (\nu )$
.
Proof Let
$g_{r}$
denote the map
$g_{r}(\alpha )=g(r,\alpha )$
and define
$\pi _{0}\colon R^{\mathbb {N}}\times E\to \Sigma $
by
$$ \begin{align*}\pi_{0}(\textbf{r},\alpha)=(\alpha,g_{r_{0}}(\alpha), \, g_{r_{1}}\circ g_{r_{0}}(\alpha)\,,\,g_{r_{2}}\circ g_{r_{1}}\circ g_{r_{0}}(\alpha)\,,\,\ldots). \end{align*} $$
Define
$\pi \colon R^{\mathbb {N}}\times E\times M\to \Sigma \times M$
by
$$ \begin{align*}\pi(\textbf{r},\alpha,x)=(\pi_{0}(\sigma(\textbf{r}),g_{r_{0}}(\alpha)),x), \end{align*} $$
where
$\sigma $
is the shift map on
$R^{\mathbb {N}}$
. It is cleat that
$\pi \circ G=F\circ \pi $
.
Now let
$\nu $
be a probability measure on
$E\times M$
with first marginal m. For every measurable set
$A\subset \Sigma \times M$
, we have
It follows from (3.2) that
where the second equality follows from
$\sigma _{*}\rho ^{\mathbb {N}}=\rho ^{\mathbb {N}}$
. Now applying (3.1), we obtain
From the definition of the Markov measure
$\mathbb {P}$
it is straightforward to check that
$$ \begin{align*} (\pi_{0})_{*}(\rho^{\mathbb{N}}\times m)=\mathbb{P}. \end{align*} $$
Therefore,
To conclude the proof, we observe that it follows from the classical correspondence in the i.i.d. case that if
$\nu $
is stationary for
$\hat p$
(respectively, ergodic), then
$\rho ^{\mathbb {N}}\times \nu $
is a G-invariant (respectively, ergodic) measure, and hence, by the Lemma 3.1,
$\pi _{*}(\rho ^{\mathbb {N}}\times \nu )=\Xi (\nu )$
is an F-invariant (respectively, ergodic) measure. This completes the proof of item (ii).
4 Upper bound for non-increasing functions on the orbits of F
In this section we state the main technical result for Markovian random iterations associated with a bounded pair
$(p,m)$
needed to prove Theorem 2.4. First, we present a result for homogeneous Markov chains similar to the Krylov–Bogolyubov theorem for continuous dynamical systems.
The following proposition can be seen as a generalization of the Breiman ergodic theorem [Reference Breiman9], and its proof is a direct consequence of Furstenberg [Reference Furstenberg15, Lemma 2.5].
Proposition 4.1. Let Y be a compact metric space and K a Feller transition probability on Y. If
$Y_{n}$
is a Markov chain with transition probability K, then, with probability
$1$
, any accumulation point (in the weak
$\star $
topology) of the sequence
$$ \begin{align*}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{Y_{i}} \end{align*} $$
is a stationary measure for K.
Proof Assume that the space of continuous functions from Y to
$\mathbb {R}$
, denoted by
$C(Y)$
, is endowed with the uniform norm topology. Since the map
$f\in C(Y)\mapsto Kf$
is non-expansive and
$C(Y)$
is separable, it follows from [Reference Furstenberg15, Lemma
$2.5$
] that, with probability
$1$
,
$$ \begin{align} \lim_{n\to \infty}\frac{1}{n}\sum_{i=0}^{n-1} Kf(Y_{i})-f(Y_{i})=0 \end{align} $$
for every
$f\in C(Y)$
. Let
$Y_{i}$
be a sample sequence for which this is the case.
Assume that
$\nu $
is an accumulation point of
$(1/n)\sum \nolimits _{i=0}^{n-1}\delta _{Y_{i}}$
, that is, there is an increasing subsequence
$(n_{k})_{k\in \mathbb {N}}$
such that
$$ \begin{align} \lim_{k\to \infty}\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\delta_{Y_{i}}= \nu \end{align} $$
in the weak
$\star $
topology. Since K is Feller, we have that
$Kf-f$
is continuous for every continuous function f. In particular, it follows from (4.2) that
$$ \begin{align*}\lim_{k\to \infty}\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}Kf(Y_{i})-f(Y_{i})= \int\! (Kf-f)\, d\nu \end{align*} $$
for every continuous function f. Now, it follows from (4.1) that
$$ \begin{align*}\int\! Kf\, d\nu=\int\! f\, d\nu \end{align*} $$
for every continuous function f. This shows that
$\nu $
is a stationary measure.
In what follows, E is a compact metric space endowed with its Borel
$\sigma $
-algebra and C denotes a constant satisfying (2.3).
Theorem 4.2. Let M be a compact metric space and consider a Markovian random iteration
$\varphi $
associated with a bounded pair
$(p,m)$
and
$f\colon E\times M\to M$
such that
$\hat p$
is Feller. Let
$F\colon \Sigma \times M\to \Sigma \times M$
be the associated skew product. Let
$\zeta \colon \Sigma \times M\to [0,\infty ]$
be a measurable map such that:
-
(i) for every
$\omega \in \Sigma $
, the map
$x\mapsto \zeta (\omega ,x)$
is lower semi-continuous; -
(ii)
$\zeta (\omega ,x)\geq \zeta (F(\omega ,x))$
for every
$(\omega ,x)\in \Sigma \times M$
.
Then, for every
$x_{0}\in M$
, for
$\mathbb {P}$
-almost every
$\omega $
there is a probability measure
$\hat \mu \in I_{0}(\varphi )$
(depending on
$\omega $
and
$x_{0}$
) such that
$$ \begin{align*}\zeta(\omega,x_{0})\geq C^{2} \int\! \zeta(\vartheta,x)\, d\hat \mu(\vartheta,x). \end{align*} $$
Remark 4.3. In this paper, we will use Theorem 4.2 only for the case where
$M=S^{1}$
and
$\zeta $
is F-invariant, that is,
$\zeta (\omega ,x)= \zeta (F(\omega ,x))$
for every
$(\omega ,x)\in \Sigma \times M$
. However, we chose to state Theorem 4.2 in such generality because the proof is the same in any case and we believe that it can be a useful tool in the study of Markovian random iterations. See [Reference Malicet24, Lemma 3.20] for the i.i.d. version of Theorem 4.2.
Before proving Theorem 4.2, we need two preliminary lemmas. We denote by
$\mathscr {F}_{n}$
the
$\sigma $
-algebra on
$\Sigma $
generated by the canonical projections
$\omega \mapsto \omega _{i}$
,
$i=0,\ldots , n$
, and by
$\mathbb {E}(\cdot |\cdot )$
the conditional expectation (with respect to
$\mathbb {P}$
) of a measurable map with respect to a
$\sigma $
-algebra.
Lemma 4.4. Let
$h\colon \Sigma \times M\to [0,\infty )$
be a bounded measurable map and
$\bar h\colon M\to [0,\infty ]$
the map defined by
$$ \begin{align*}\bar h(x)=\int\! h(\omega,x)\, d\mathbb{P}(\omega). \end{align*} $$
Then, for every
$n\geq 1$
and
$x_{0}\in M$
, we have
$$ \begin{align*}\mathbb{E}(h(F^{n}(\cdot, x_{0}))|\mathscr{F}_{n-1}) \geq C \bar h(f_{\omega}^{n}(x_{0})) \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega $
.
Proof By the definition of conditional expectations, we need to prove that, for every measurable set
$D\in \mathscr {F}_{n-1}$
,
$$ \begin{align} \int_{D}h(\sigma^{n}(\omega),f^{n}_{\omega}(x_{0}))\,d\mathbb{P}(\omega)\geq C\int_{D}\int\! h(\vartheta, f_{\omega}^{n}(x_{0}))\, d\mathbb{P}(\vartheta)\, d\mathbb{P}(\omega). \end{align} $$
We start by proving that (4.3) holds for every characteristic function
, where
$A\times B\in \mathscr {F}\otimes \mathscr {B}$
. Note that
. Hence, we have
Recall that if
$u\colon \Sigma \to [0,\infty )$
is a measurable map and
$g\colon \Sigma \to [0,\infty )$
is a
$\mathscr {F}_{n}$
-measurable map, then the Markov property (see [Reference Revuz27, Proposition
$2.13$
]) implies that
$$ \begin{align} \int\! (u\circ \sigma^{n})g\, d\mathbb{P}=\int\! \mathbb{E}_{\omega_{n}}(u) g(\omega)\, d\mathbb{P}(\omega), \end{align} $$
where
$\mathbb {E}_{\alpha }$
denotes the expectation with respect to the Markov measure defined by the transition probability p and starting measure
$\delta _{\alpha }$
. Applying (4.5) on the right-hand side of (4.4) with
and
(g is
$\mathscr {F}_{n-1}$
-measurable, and in particular, it is
$\mathscr {F}_{n}$
-measurable), we obtain
Since the mapping
is
$\mathscr {F}_{n}$
-measurable, it follows from the definition of the Markov measure
$\mathbb {P}$
that
Since the mapping
does not depend on
$\alpha $
, we have
Recalling that by hypothesis
$$ \begin{align*}C\leq \frac{dp_{\alpha}}{dm}\leq C^{-1} \end{align*} $$
for every
$\alpha \in E$
, we conclude that
This shows that (4.3) holds for
.
We now observe that if R is an element of the algebra generated by the measurable rectangles, then the map
satisfies (4.3). It is readily checked that the class of measurable sets R of
$\Sigma \times M$
for which the map
satisfies (4.3) is a monotone class. Since the class of measurable rectangles generates the product
$\sigma $
-algebra of
$\Sigma \times M$
, it follows from the monotone class theorem that if R is a subset of
$\mathscr {F}\otimes \mathscr {B}$
, then the map
satisfies (4.3). Now, by standard arguments of measure theory, it is easily verified that every non-negative measurable map satisfies (4.3). This completes the proof of the lemma.
For the next lemma, denote by
$\Pi _{2}\colon E\times M\to M$
the projection on the second factor given by
$\Pi _{2}(\alpha ,x)=x$
.
Lemma 4.5. Let
$\Xi \colon S(\varphi )\to I_{0}(\varphi )$
be the bijection given by item (ii) of Theorem 2.1. For any
$\nu \in S(\varphi )$
, we have
$$ \begin{align*}C \Xi(\nu)\leq \mathbb{P}\times \Pi_{2*}\nu. \end{align*} $$
Proof It is enough to prove that, for every measurable rectangle
$A\times B\in \mathscr {F}\otimes \mathscr {B}$
, we have
$$ \begin{align*}C \Xi(\nu)(A\times B)\leq \mathbb{P}\times \Pi_{2*}\nu (A\times B). \end{align*} $$
Fix
$A\times B\in \mathscr {F}\otimes \mathscr {B}$
. We claim that
$$ \begin{align*}\Pi_{2*}\nu (B)\geq C\int\! \nu_{\beta}(B)\, q(\omega_{0},d\beta) \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega $
(or equivalently, for m-almost every
$\omega _{0}$
). To prove this claim, consider a measurable set
$D\subset E$
with
$m(D)>0$
. Using that
$(p,m)$
is bounded and the relation (3.1), we obtain
where the last equality follows from the definition of disintegration of measures. Since D is an arbitrary measurable set with
$m(D)>0$
, we conclude from [Reference Rudin29, Theorem 1.40] that, for
$\mathbb {P}$
-almost every
$\omega $
,
$$ \begin{align*}\int\! \nu_{\beta}(B)\, q(\omega_{0},d\beta)\leq C^{-1}\Pi_{2*}\nu(B). \end{align*} $$
Therefore, by the definition of
$\Xi (\nu )$
, we have
$$ \begin{align*}\mathbb{P}\times \Pi_{2*}\nu(A\times B)=\int_{A}\Pi_{2*}\nu(B)\, d\mathbb{P}(\omega)\geq C\int_{A} \hat \Xi(\nu)_{\omega}(B)\, d\mathbb{P}(\omega)=C\hat \Xi(\nu)(A\times B), \end{align*} $$
which is the desired result.
Proof of Theorem 4.2
Denote by
$A_{\omega ,x_{0}}$
the set of accumulation points of the sequence of probability measures
$({1}/{n})\sum \nolimits _{i=0}^{n-1}\delta _{f_{\omega }^{i}(x_{0})}$
. We need the following estimate.
Proposition 4.6. For every
$x_{0}\in M$
, for
$\mathbb {P}$
-almost every
$\omega $
, we have
$$ \begin{align*}\zeta(\omega,x_{0})\geq C \mathop{\mathrm{sup}}\limits_{\mu\in A_{\omega,x_{0}}} \int\! \int\! \zeta(\vartheta,x)\,d\mathbb{P}(\vartheta)\, d\mu(x). \end{align*} $$
Proof We first prove the proposition in the case where
$\zeta $
is bounded. Given
$\mu \in A_{\omega ,x_{0}}$
, by definition there is a subsequence
$n_{k}$
such that
$$ \begin{align*}\lim_{k\to \infty} \frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\delta_{f_{\omega}^{i}(x_{0})}=\mu. \end{align*} $$
Let
$\mathscr {F}_{n}$
be the
$\sigma $
-algebra generated by the canonical projections
$\omega \mapsto \omega _{i}$
,
$i=0,\ldots , n$
. From Levy’s
$0$
–
$1$
law we have that
$\zeta (\omega ,x_{0})=\lim _{n\to \infty } \mathbb {E}(\zeta (\cdot ,x_{0})|\mathscr {F}_{n})(\omega )$
for
$\mathbb {P}$
-almost every
$\omega \in \Sigma $
. In particular,
$$ \begin{align} \zeta(\omega,x_{0})=\lim_{k\to \infty} \frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1} \mathbb{E}(\zeta(\cdot,x_{0})|\mathscr{F}_{i})(\omega) \end{align} $$
for
$\mathbb {P}$
-almost every
$\omega $
. Since
$\zeta (\omega ,x)\geq \zeta (F(\omega ,x))$
for every
$(\omega ,x)\in \Sigma \times M$
, we have
$\zeta (\omega ,x_{0})\geq \zeta (F^{i}(\omega ,x_{0}))$
for every
$\omega $
and i, which implies that
$$ \begin{align} \mathbb{E}(\zeta(\cdot,x_{0})|\mathscr{F}_{i-1})(\omega)\geq\mathbb{E}(\zeta(F^{i}(\cdot,x_{0}))|\mathscr{F}_{i-1})(\omega). \end{align} $$
Then it follows from (4.7), (4.6) and Lemma 4.4 that, for
$\mathbb {P}$
-almost every
$\omega $
, we have
$$ \begin{align*} \zeta(\omega,x_{0})&\geq C \liminf_{k\to \infty} \frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1} \bar \zeta(f_{\omega}^{n}(x_{0}))\\ &=C\liminf_{k\to \infty} \int\! \bar\zeta \, d\bigg( \frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1} \delta_{f_{\omega}^{n}(x_{0})} \bigg). \end{align*} $$
Next, because
$\zeta $
is lower semi-continuous, the map
$\bar \zeta $
is also lower semi-continuous. Hence, it follows from the portmanteau theorem that
$$ \begin{align*}\zeta(\omega,x_{0})\geq C \int\! \bar \zeta \, d\mu=C\int\! \int\! \zeta(\vartheta,x)\,d\mathbb{P}(\vartheta)d\mu(x) \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega $
. This completes the proof of the proposition in the case where
$\zeta $
is bounded. For a general non-negative measurable function
$\zeta $
, the proposition is obtained through the cut-off functions
$\min \{N, \zeta \}$
and the monotone convergence theorem.
We are now ready to conclude the proof of the theorem. Fix
$x_{0}\in M$
and recall that
$Z_{n}(\omega )=(\omega _{n-1},f_{\omega }^{n}(x_{0}))\in E\times M$
is a Markov chain with transition probability
$\hat p$
. Let
$S_{\omega ,x_{0}}$
be the set of accumulation points of the sequence of probability measures
$({1}/{n})\sum \nolimits _{i=0}^{n-1}\delta _{Z_{i}(\omega )}$
.
Since
$E\times M$
is a compact metric space and
$\hat p$
is Feller, it follows from Proposition 4.1 that there is a subset
$\Sigma _{0}\subset \Sigma $
of
$\mathbb {P}$
-full measure such that, for every
$\omega \in \Sigma _{0}$
, the set
$S_{\omega ,x_{0}}$
is constituted by stationary measures for
$\varphi $
, that is,
$S_{\omega ,x_{0}}\subset S(\varphi )$
. We also can assume that, for every
$\omega \in \Sigma _{0}$
, Proposition 4.6 holds true for
$(\omega ,x_{0})$
.
Let
$\Xi \colon S(\varphi )\to I_{0}(\varphi )$
be the bijection of Theorem 2.1. Then, for every
$\omega \in \Sigma _{0}$
, take any
$\nu \in S_{\omega ,x_{0}}$
and let
$\hat \mu =\Xi (\nu )$
. By the definition of
$S_{\omega ,x_{0}}$
, there is a subsequence
$(n_{k})_{k}$
such that
$$ \begin{align*}\nu=\lim_{n\to \infty}\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\delta_{(\omega_{i-1},f_{\omega}^{i}(x_{0}))}. \end{align*} $$
Let
$\Pi _{2}$
be the projection on the second factor of
$E\times M$
. We observe that
$\Pi _{2*}\nu \in A_{\omega ,x_{0}}$
, and hence we can apply Proposition 4.6 to obtain
$$ \begin{align} \zeta(\omega,x_{0})\geq C\iint \zeta(\vartheta,x) \, d\Pi_{2*}\nu(x) d\mathbb{P}(\vartheta). \end{align} $$
We now apply Lemma 4.5 to equation (4.8) to obtain
$$ \begin{align*} \zeta(\omega,x_{0}) &\geq C^{2} \int\! \zeta(\vartheta,x) \, d\hat \mu(\vartheta,x). \end{align*} $$
The proof of the theorem is now complete.
5 Local synchronization and proof of Theorem 2.4
In this section we prove Theorem 2.4. We start by presenting a reformulation of Theorem 2.4 using the exponent of contraction of a random iteration of homeomorphisms introduced in [Reference Malicet24].
5.1 Reformulation of Theorem 2.4
Let
$\varphi $
be a Markovian random iteration of homeomorphisms of the circle
$S^{1}$
. The exponent of contraction of
$\varphi $
at the point
$(\omega ,x)\in \Sigma \times S^{1}$
is the non-positive quantity
$$ \begin{align*}\lambda(\omega,x)=\varlimsup_{y\to x}\, \varlimsup_{n\to \infty} \frac{\log\, \mathrm{Leb}\,f^{n}_{\omega}([x,y])}{n} \end{align*} $$
where
$\mathrm {Leb}$
denotes the Lebesgue measure on
$S^{1}$
and
$[a,b]$
denotes the arc of smaller length determined by the points a and b.
Theorem 5.1. Under the assumptions of Theorem 2.4, there is
$\lambda _{0}<0$
such that, for every
$x\in S^{1}$
, we have
$$ \begin{align*}\lambda(\omega,x)\leq \lambda_{0} \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega $
in
$\Sigma $
.
The proof of Theorem 5.1 is a consequence of Theorem 4.2 combined with the invariance principle [Reference Malicet24, Theorem F]. For the precise statement of the invariance principle for
$\varphi $
, we need the following definition. Let
$\hat \mu $
be a
$\varphi $
-invariant measure. The exponent of contraction of
$\hat \mu $
is the non-positive quantity
$$ \begin{align*}\lambda(\hat\mu)=\int\!\lambda(\omega,x) \, d\hat \mu(\omega,x). \end{align*} $$
Theorem 5.2. (Invariance principle)
Let
$\hat \mu $
be a
$\varphi $
-invariant measure such that
$\lambda (\hat \mu )=0$
. Then
$$ \begin{align*}f_{\omega_{0}*}\hat \mu_{\omega}=\hat \mu_{\sigma(\omega)} \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega \in \Sigma $
.
Theorem 5.2 is just a particular case of a general result proved by Malicet in [Reference Malicet24, Theorem F] for stationary random iterations of homeomorphisms of
$S^{1}$
.
Proof of Theorem 5.1
We start by showing that, for every
$\hat \mu \in I_{0}(\varphi ) $
, we have
$\lambda (\hat \mu )<0$
. Indeed, assume that
$\lambda (\hat \mu )=0$
. Applying Theorem 5.2, we obtain
$$ \begin{align} f_{\omega_{0}*}\hat \mu_{\omega}=\hat \mu_{\sigma(\omega)} \end{align} $$
for
$\mathbb {P}$
-almost every
$\omega $
. Let
$\{\mu _{\alpha }\colon \alpha \in E\}$
be the family of probability measures for which
$\hat \mu _{\omega }=\mu _{\omega _{0}}$
. Hence, it follows from (5.1) that
$$ \begin{align*}f_{\omega_{0}*}\mu_{\omega_{0}}=\mu_{\omega_{1}} \end{align*} $$
for
$\mathbb {P}$
-almost every
$\omega $
, or equivalently, for m-almost every
$\alpha $
, we have
$$ \begin{align} f_{\alpha*}\mu_{\alpha}=\mu_{\beta} \end{align} $$
for
$p_{\alpha }$
-almost every
$\beta $
. Because
$m\ll p_{\alpha }$
for m-almost every
$\alpha $
, we get from (5.2) that
$\beta \mapsto \mu _{\beta }$
is constant (m-almost everywhere). This shows that the maps
$f_{\alpha }$
have an invariant measure in common, which is a contradiction. Therefore, the exponent of contraction
$\lambda (\hat \mu )$
is negative.
We now observe that
$\lambda $
is F-invariant and, for every
$\omega $
, the map
$x\mapsto \lambda (\omega ,x)$
is upper semi-continuous. In particular,
$-\lambda $
is F-invariant and, for every
$\omega $
, the map
$x\mapsto -\lambda (\omega ,x)$
is lower semi-continuous. Hence, all conditions required in Theorem 4.2 are met for the map
$-\lambda $
, and so we can apply this theorem to get that, for every
$x\in S^{1}$
, for
$\mathbb {P}$
-almost every
$\omega $
, there exists
$\hat \mu \in I_{0}(\varphi )$
such that
$$ \begin{align*}\lambda(\omega,x)\leq C^{2}\lambda(\hat \mu). \end{align*} $$
Since
$\lambda (\hat \mu )<0$
for every
$\hat \mu \in I_{0}(\varphi )$
, we conclude that, for every
$x\in S^{1}$
, for
$\mathbb {P}$
-almost every
$\omega $
, we have
$$ \begin{align} \lambda(\omega,x)< 0. \end{align} $$
It remains to see the existence of a uniform bound as claimed in Theorem 5.1. To this end, we apply Proposition 4.6 to the map
$-\lambda $
to obtain that, for every
$x_{0}\in M$
, for
$\mathbb {P}$
-almost every
$\omega $
, we have
$$ \begin{align} \lambda(\omega,x_{0})\leq C \inf_{\mu\in A_{\omega,x_{0}}} \int\! \int\! \lambda(\vartheta,x)\,d\mathbb{P}(\vartheta)\, d\mu(x), \end{align} $$
where
$A_{\omega ,x_{0}}$
is the set of accumulation points of the sequence of probability measures
$({1}/{n})\sum \nolimits _{i=0}^{n-1}\delta _{f_{\omega }^{i}(x_{0})}$
. We now use the fact that
$\lambda $
is upper semi-continuous on the second variable to conclude that the map
$\bar \lambda \colon S^{1}\to \mathbb {R}$
defined by
$$ \begin{align*}\bar\lambda(x)= \int\! \lambda(\omega,x)\, d\mathbb{P}(\omega) \end{align*} $$
is also upper semi-continuous. Because of (5.3), we have that
$\bar \lambda (x)<0$
for every
$x\in S^{1}$
. Since any upper semi-continuous map on a compact metric space has a maximum value, we conclude that there is
$\lambda _{0}<0$
such that
$$ \begin{align*}\bar\lambda(x)\leq \lambda_{0} \end{align*} $$
for every
$x\in S^{1}$
. Therefore, because
$C\leq 1$
, it follows from (5.4) that, for every
$x_{0}$
, for
$\mathbb {P}$
-almost every
$\omega $
, we have
$$ \begin{align*}\lambda(\omega,x_{0})\leq \inf_{\mu\in A_{\omega,x_{0}}} \int\! \bar \lambda(x)\, d\mu(x)\leq \lambda_{0}, \end{align*} $$
which is the desired result.
6 Related and future work
A natural question would be whether we can obtain generalizations of Theorem 2.4 for non-Markovian random iterations of homeomorphisms of
$S^{1}$
. This question leads us to a general question on ergodic theory. Indeed, recall that the proof of Theorem 2.4 requires a Krylov–Bogolyubov-type result for homogeneous Markov chains; see Proposition 4.1. Therefore, it is reasonable to require a version of this result for skew products.
To be more precise, we need some definitions. Let
$(\Omega ,\mathscr {F},\mathbb {P},\theta )$
be an invertible measure-preserving dynamical system. Consider a measurable space M and let
$f\colon \Omega \times M\to M$
be a measurable map. Let
$f_{\omega }$
be the map given by
$f_{\omega }(x)=f(\omega ,x)$
. Then the map
$\varphi \colon \mathbb {N}\times \Omega \times M \to M$
defined by
$$ \begin{align*}\varphi(n,\omega,x)=f_{\theta^{n-1}(\omega)}\circ \cdots\circ f_{\omega}(x) \end{align*} $$
is called a (stationary) random iteration of maps. A probability measure
$\mu $
on
$\Omega \times M$
is called
$\varphi $
-invariant if
$\mu $
has first marginal
$\mathbb {P}$
and is invariant by the skew product map F defined by
$$ \begin{align*}F(\omega,x)=(\theta(\omega),f_{\omega}(x)). \end{align*} $$
The past of
$\varphi $
is the
$\sigma $
-algebra defined by
$$ \begin{align*}\mathscr{F}^{-}\stackrel{\scriptscriptstyle\rm def}{=}\sigma(\omega\mapsto f_{\theta^{-n}(\omega)}^{r}(x)\colon n\geq r\geq 1 , x\in X), \end{align*} $$
that is,
$\mathscr {F}^{-}$
is the smallest
$\sigma $
-algebra that makes all maps in the above family measurable. A probability measure
$\mu $
on
$ \Omega \times M$
with first marginal
$\mathbb {P}$
is called a forward Markov measure if, for every measurable set A, the map
$\omega \mapsto \mu _{\omega }(A)$
is
$\mathscr {F}^{-}$
-measurable, where
$\{\mu _{\omega }\colon \omega \in \Omega \}$
is the disintegration of
$\mu $
with respect to
$\mathbb {P}$
. This notion was introduced by Crauel in [Reference Crauel11]; see also [Reference Arnold5, Reference Crauel12].
We recall that if
$\Omega $
is a product space
$E^{\mathbb {Z}}$
,
$\theta $
is the shift map and
$\mathbb {P}$
is a product measure
$\nu ^{\mathbb {Z}}$
, then there is a one-to-one correspondence between the set of stationary measures and the set of
$\varphi $
-invariant forward Markov measures; see Arnold [Reference Arnold5, Theorem 2.1.8]. Note that Theorem 2.1 combined with [Reference Arnold5, Theorem 1.7.2] implies that there is such a correspondence also in the Markovian case.
Thus, when we cannot consider stationary measures, it seems natural to take the set of
$\varphi $
-invariant forward Markov measure to generalize results on which stationary measures play a role. With these remarks in mind, we pose the following question.
Question 1. Assume that
$\Omega $
and M are compact metric spaces endowed with their Borel
$\sigma $
-algebras and F is a continuous map. For every x, for
$\mathbb {P}$
-almost every
$\omega $
, is every accumulation point (in the weak
$\star $
topology) of
$$ \begin{align*}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{F^{i}(\omega,x)} \end{align*} $$
a
$\varphi $
-invariant forward Markov measure?
Acknowledgments
The author thanks the anonymous referee for a careful reading of the paper and constructive suggestions, especially for providing a much shorter proof of item (ii) of Theorem 2.1, which also allowed the correspondence between ergodic measures to be stated in greater generality. This work was partially completed while the author was supported by postdoctoral fellowships at Federal University of Rio de Janeiro and ICMC-USP. The author thanks CAPES and Serrapilheira for the financial support. The author warmly thanks Anna Zdunik, Tiago Pereira and Katrin Gelfert for their useful comments. The author is also very grateful to Katrin Gelfert and Tiago Pereira for their generous hospitality.
