Extremal exponents of random products of conservative diffeomorphisms

Abstract

We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\ge 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of independent conservative surface diffeomorphisms.

Apendix A: Stationary measure

Let us consider two standard Borel probability spaces $(X,\mu )$ and $(\Omega ,{\mathbb {P}})$ and endow the product space ${\bar{X}}=\Omega \times X$ with the product measure $\bar{\mu }={\mathbb {P}}\times \mu$. Consider a $\bar{\mu }$-preserving measurable skew-product map

$$\begin{aligned} {\bar{f}}: {\bar{X}}\rightarrow {\bar{X}}, \quad {\bar{f}}(\omega ,x)=(\theta \omega ,f_\omega (x)) \end{aligned}$$

where $\theta$ is a ergodic ${\mathbb {P}}$-invariant invertible continuous transformation of $\Omega$ and $f_\omega :X\rightarrow X$ are continuous $\mu$-preserving maps for ${\mathbb {P}}$-almost all $\omega \in \Omega$. Let Z be a locally compact Hausdorff topological space and consider the induce Borel $\sigma$-algebra. Again, consider a skew-product map

$$\begin{aligned} F:{\bar{X}}\times Z \rightarrow {\bar{X}}\times Z, \quad F({\bar{x}},z)=({\bar{f}}({\bar{x}}), A({\bar{x}})z) \end{aligned}$$

(A.1)

where $A({\bar{x}}):Z\rightarrow Z$ are continuous maps $\mu$-almost surely. Sometimes we write $F={\bar{f}}\ltimes A({{\bar{x}}})$ or $F=({\bar{f}},A)$ when no confusion can arise to emphasize the base map and the fiber maps of F. Observe that since ${\bar{f}}=\theta \ltimes f_\omega$ is also a skew-product map, we can also rewrite $F= \theta \ltimes F_\omega$ where $F_\omega = f_\omega \ltimes A(\omega ,x)$. That is, $F= \theta \ltimes f_\omega \ltimes A(\omega ,x)$.

Definition A.1

A measure ${\hat{\nu }}$ on $X\times Z$ is $({\bar{f}},A)$-stationary if ${\hat{\nu }}$ projets down on X over $\mu$ and

$$\begin{aligned} {\hat{\nu }} = \int F_\omega {\hat{\nu }} \, d{\mathbb {P}}(\omega ). \end{aligned}$$

Notice that by definition a stationary measure is not necessarily a probability measure. Denote by ${\mathscr {M}}(Z)$ be the Banach space of signed finite (not necessarily probability) Borel measures on Z with variation norm. This Banach space can be identified with the dual of $C_0(Z)$, the space of bounded continuous real-valued functions on Z vanishing at infinity with supremum norm. Then we can endow ${\mathscr {M}}(Z)$ with the weak$^*$ topology and consider the Borel $\sigma$-algebra induced by this topology. Let $L^\infty (X;{\mathscr {M}}(Z))$ be the Banach space of (equivalence classes of) essentially bounded measurable mappings from X to ${\mathscr {M}}(Z)$. Given $\nu : x \mapsto \nu _x$ in $L^\infty (X;{\mathscr {M}}(Z))$ define the measure ${\hat{\nu }}$ on $X\times Z$ by

$$\begin{aligned} {\hat{\nu }}(E\times B) = \int _E \nu _x(B) \, d\mu (x) \end{aligned}$$

for E, B measurable sets on X and Z respectively and extending to the product $\sigma$-algebra. By definition ${\hat{\nu }}$ has as marginal $\mu$ and disintegration $\nu : x \mapsto \nu _x$. That is, $d{\hat{\nu }}= \nu _x\,d\mu (x)$.

Notice that $L^\infty (X;{\mathscr {M}}(Z))$ can be identified with the dual of the Banach space

$$\begin{aligned} L^1(X;C_0(Z)){\mathop {=}\limits ^{\scriptscriptstyle \mathrm{def}}}\{h:X \rightarrow C_0(Z): \ \Vert h_x\Vert _\infty \in L^1(X,\mu ) \}. \end{aligned}$$

We introduce the transfer operator ${\mathscr {P}}$ on $L^1(X;C_0(Z))$ defined by

$$\begin{aligned} {\mathscr {P}}\varphi {\mathop {=}\limits ^{\scriptscriptstyle \mathrm{def}}}\int \varphi \circ F_\omega \, d{\mathbb {P}}(\omega ) \in L^1(X;C_0(Z)) \ \ \text {for }\varphi \in L^1(X;C_0(Z)). \end{aligned}$$

It is clear that ${\mathscr {P}}$ is a bounded linear operator. The adjoint transition operator ${\mathscr {P}}^*$ acts on $L^\infty (X;{\mathscr {M}}(Z))$ by taking $\nu : x \mapsto \nu _x$ in $L^\infty (X;{\mathscr {M}}(Z))$ and defining ${\mathscr {P}}^*\nu \in L^\infty (X;{\mathscr {M}}(Z))$ as

$$\begin{aligned} {\mathscr {P}}^*\nu : x \mapsto ({\mathscr {P}}^*\nu )_x=\int A(\theta ^{-1}\omega ,f_\omega ^{-1}(x)) \nu _{f_\omega ^{-1}(x)} \,d{\mathbb {P}}(\omega ). \end{aligned}$$

Consequently ${\mathscr {P}}^*$ is also a bounded linear operator. Moreover, if ${\mathscr {P}}^*\nu =\nu$, then ${\hat{\nu }}$ is a $({\bar{f}},A)$-stationary measure on $X\times Z$. Indeed,

$$\begin{aligned} \int F_\omega {\hat{\nu }}(E\times B) \, d{\mathbb {P}}(\omega )&= \int \int _E A(\theta ^{-1}\omega ,f_\omega ^{-1}(x)) \nu _{f_\omega ^{-1}(x)}(B) \, d\mu (x)d{\mathbb {P}}(\omega ) \\&=\int _E ({\mathscr {P}}^*\nu )_x(B) \, d\mu (x)=\int _E \nu _x(B)\,d\mu (x) = {\hat{\nu }}(E\times B) \end{aligned}$$

for all E, B measurable sets on X and Z respectively. Hence ${\hat{\nu }}$ is a $({\bar{f}},A)$-stationary measure.

According to Banach-Alaoglu’s theorem the unit ball in $L^\infty (X;{\mathscr {M}}(Z))$ is a compact set. Moreover, since X is a standard probability space, its $\sigma$-algebra is countably generated. This implies that $L^1(X;C_0(Z))$ is separable and thus $L^\infty (X;{\mathscr {M}}(Z))$ is metrizable [8]. Consequently the unit ball in $L^\infty (X;{\mathscr {M}}(Z))$ is also sequentially compact. Now, the existence of stationary probability measures for random cocycles follows from standard arguments when Z is compact.

Proposition A.2

Let $({\bar{f}},A)$ be skew-product as (A.1) acting on ${\bar{X}}\times Z$. If Z is a compact Hausdorff topological space, then the set of $({\bar{f}},A)$-stationary probability measures on $X\times Z$ is nonvoid.

Proof

Denote by ${\mathscr {P}}(Z)$ the subset of ${\mathscr {M}}(Z)$ of probability measure. Notice that

$$\begin{aligned} L^\infty (X;{\mathscr {P}}(Z))=\{\nu \in L^\infty (X;{\mathscr {P}}(Z)): \nu _x \in {\mathscr {P}}(Z) \ \ \nu \text {-almost surely} \} \end{aligned}$$

is a convex subset of the unit ball in $L^\infty (X;{\mathscr {M}}(Z))$. Moreover, ${\mathscr {P}}^*$ leaves $L^\infty (X;{\mathscr {P}}(Z))$ invariant. Since, by assumption Z is compact, $L^\infty (X;{\mathscr {P}}(Z))$ is closed and hence compact in the weak$^*$ topology. Brouwer’s fixed-point theorem yields the existence of a ${\mathscr {P}}^*$-invariant element $\nu \in L^\infty (X;{\mathscr {P}}(Z))$. This yields a $({\bar{f}},A)$-stationary measure ${\hat{\nu }}$ on $X\times Z$ defined by $d{\hat{\nu }}=\nu _x\,d\mu (x)$ and completes the proof. $\square$

When Z is not compact we can not guarantee in general that the set of $({\bar{f}},A)$-stationary probability measure is nonvoid. However the following lemma provides a powerful method to find stationary finite measure.

Lemma A.3

Let $\nu \in L^\infty (X;{\mathscr {P}}(Z))$. Then, the set of accumulation point $\eta \in L^\infty (X;{\mathscr {M}}(Z))$ of the sequence $(\nu _n)_n$ given by

$$\begin{aligned} \nu _n=\frac{1}{n}\sum _{j=0}^{n-1} {\mathscr {P}}^{*j}\nu \in L^\infty (X;{\mathscr {P}}(Z)) \quad \text {for }n\in {\mathbb {N}} \end{aligned}$$

is nonvoid. Moreover, any accumulation point $\eta$ of $(\nu _n)_n$ defines a $({\bar{f}},A)$-stationary finite measure ${\hat{\eta }}$ on $X\times Z$ whose disintegration is $\eta$. Consequently ${\hat{\eta }}$ is an accumulation point in the weak$^*$ topology of the sequence of probability measures $({\hat{\nu _n}})_n$ on $X\times Z$ defined by the disintegrations $(\nu _n)_n$.

Proof

Since the unit ball of $L^\infty (X;{\mathscr {M}}(Z))$ is sequentially compact then we can extract a convergent subsequence from $(\nu _n)_n$ and thus the set of accumulation points is not empty. Moreover, any accumulation point belongs to this ball. Thus $(\nu _n)_x$ is a finite measure $\mu$-almost surly. On the other hand, by well known arguments the limit $\eta$ of any convergent sequences of $(\nu _n)_n$ is also ${\mathscr {P}}^*$-invariant and thus ${\hat{\eta }}$ is a $({\bar{f}},A)$-stationary measure on $X\times Z$. $\square$