Absolutely continuous invariant measures for random non-uniformly expanding maps

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps.

1. The equicontinuity can be replaced by the following condition: given \(\epsilon >0\), there exists \(\delta >0\) such that if \(|x-{\fancyscript{C}}_\theta |<\delta \) then \(|f'_{\theta }(x)|<\epsilon \), for all \(\theta \in \mathbb {X}\). This is used in the proof of Theorem 4.1.

The authors thank Universidade Federal do Rio de Janeiro (UFRJ) and IMPA, at Rio de Janeiro, Brasil, and also Pontificia Universidade Catolica de Valparaiso (PUCV), at Valparaiso, Chile, where part of this work was developed, for their hospitality. We thank the anonymous referee for the detailed suggestions that helped improved the presentation and the readability of the text.

Authors and Affiliations

1. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador , 40170-110, Brazil

   Vitor Araujo

2. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga, s/n, Valonguinho, Niterói-RJ , 24.020-140, Brazil

   Javier Solano

Corresponding author

Correspondence to Vitor Araujo.

V. Araujo and J. Solano were partially supported by CNPq, FAPERJ, FAPESB and PRONEX (Brazil).

Appendix: Measurability

Here we prove that the measures \(\eta _n\) defined on Sect. 3 together with the measures \(\mu _n\) defined on Sect. 4 are well-defined. We consider separately the case with one dimensional fibers and the case with higher dimensional fibers.

1.1 The measures \(\eta _n\) are well defined

By the Hahn Extension Theorem, it is enough to define the measures on rectangles \(A\times J\) with \(A\in \mathcal {B}_{\mathbb {X}}\) and \(J\in \mathcal {B}_{I_0}\). It easily follows from

Proposition 9.1

Let \(J\subset I_0\) be a Borel set. For every \(n\in \mathbb {N}\), the function \(\mathbb {X}\ni \theta \mapsto \eta _n(\theta )(J)\) is measurable.

Proof of Lemma 8.4

Let us fix a set \(J\in \mathcal {B}_{I_0}\). To prove the measurability of \(\theta \mapsto \eta _n(\theta )(J)\) it suffices to prove the measurability of the functions \(\theta \mapsto \eta ^{i}_{J}(\theta )\,{:=}\,(f_{\alpha ^{-i}(\theta )}^{i})_{*} \mathrm{m }(J)\), for \(i\in \mathbb {N}\). Let us define the following functions

and \(\chi _J\) is the characteristic function of \(J\). The projection maps are clearly measurable, considering on \(\mathbb {X}\times I_0\) the \(\sigma \)-algebra \(\mathcal {B}_{\mathbb {X}}\times \mathcal {B}_{I_0}\). Since compositions of measurable maps are measurable maps, \(\alpha ^{-1}\times id (\theta ,x)=(\alpha ^{-1}\circ \pi _{\mathbb {X}}(\theta ,x), \pi _{I_0}(\theta ,x))\) is also measurable.

With these notations, we have that \(\eta ^{i}_{J}(\theta )= \int _{I_0} \phi _i(\theta ,x)\,d\mathrm{m }(x)\), where \(\phi _i:\mathbb {X}\times I_0\rightarrow \mathbb {R}\) is defined by

Using Fubini’s Theorem, the measurability of \(\phi _i\) (considering the \(\sigma \)-algebra \(\mathcal {B}_{\mathbb {X}}\times \mathcal {B}_{I_0}\)) implies the measurability of \(\theta \mapsto \eta ^{i}_{J}(\theta )\) (considering the \(\sigma \)-algebra \(\mathcal {B}_{\mathbb {X}}\)) .\(\square \)

1.2 The measures \(\mu _n\) are well defined

We assume the skew-product satisfies the property \((H_4)\). The proof for the case of \((H_4^*)\) is entirely analogous. It is enough to substitute \({\fancyscript{C}}\) by \({\fancyscript{D}}\).

As in the case of \(\eta _n\), the well-definition of the measures \(\mu _n\) follows from Hahn Extension Theorem and the following result which implies that these measures are defined on the algebra of the rectangles.

Proposition 9.2

Let \(J\subset I_0\) be a borelian set. For every \(n\in \mathbb {N}\), the function \(\mathbb {X}\ni \theta \mapsto \mu _n(\theta )(J)\) is measurable.

In the definition of the measures \(\mu _n\) appear the sets \(H_j(\theta ,\varsigma )\) (\(j\in \mathbb {N}, \theta \in \mathbb {X}\)). These sets depend on the maps \(r_j(\theta ,x)\) and \(l^{*}_{j}(\theta ,x)\,{:=}\, |f_{\theta }^{j}(T^j(\theta ,x))|\). We study first the measurability of these functions.

Let us recall the definition of the function \(r_i\) (given in Sect. 4). Given \(i\in \mathbb {N}\) and a point \((\theta , x)\in \mathbb {X}\times I_0\), we denote by \(T_i(\theta ,x)\) the maximal interval such that \(f_{\theta }^{j}(T_i(\theta ,x))\cap {\fancyscript{C}}_{\alpha ^j(\theta )}=\emptyset \) for all \(j<i\). Thus \(r_i(\theta ,x)\) denotes the minimum of the lengths of the connected components of \(f_{\theta }^{i}(T_i(\theta ,x)\setminus \{x\})\).

Lemma 9.3

The maps \(r_i:\mathbb {X}\times I_0 \rightarrow \mathbb {R}\) are measurable, for all \(i\in \mathbb {N}\).

Proof of Lemma 8.4

For fixed \(\theta \in \mathbb {X}\), \(x\mapsto r_i(\theta ,x)\) is a continuous function, since \(f_\theta ^{i}\) (for \(\theta \in \mathbb {X}\), \(i\in \mathbb {N}\)) are piecewise continuous \(C^3\) maps. Hence, by [23, Lemma 9.2], we conclude \(r_i\) is measurable, if for fixed \(x\in I_0\) the function \(\theta \mapsto r_i(\theta ,x)\) is measurable. We claim that this last condition is true. To prove it, we write \(r_i(\cdot , x)\) as a composition of measurable maps.

For \(i\in \mathbb {N}\), let us define the set

Given \((\theta ,x)\in \mathbb {X}\times I_0\), the interval \(T_i(\theta ,x)=(a_i(\theta ,x),b_i(\theta ,x))\) can be defined in the following way

where \(E^{x-}= (\mathbb {X}\times (-\infty ,x]\cap I_0)\cap {\fancyscript{C}}^{i}\) and \(E^{x+}= (\mathbb {X}\times [x,+\infty )\cap I_0)\cap {\fancyscript{C}}^{i}\). The sets \(E^{x-}\) and \(E^{x+}\) are measurable, since by hypotheses \((H_1)\), \({\fancyscript{C}}\) is measurable. Then, for fixed \(x\in I_0\), the measurability of the functions \(\theta \mapsto a_i(\theta ,x)\) and \(\theta \mapsto b_i(\theta ,x)\) follows from the next result.\(\square \)

Claim 9.4

Let \(E\) be a set in \(\mathcal {B}_{\mathbb {X}}\times \mathcal {B}_{I_0}\) and let \(S:\mathbb {X}\rightarrow I_0, s:\mathbb {X}\rightarrow I_0\) be functions defined by \(S(\theta )=\sup E_{\theta }=\sup \{y\in I_0; (\theta ,y)\in E\}, s(\theta )=\inf E_{\theta }=\inf \{y\in I_0; (\theta ,y)\in E\}\). Then \(S\) and \(s\) are measurable maps.

Proof of Lemma 8.4

We prove first for the map \(S\). Let \(b\in \mathbb {R}\) be a constant. We want to prove that \(S^{-1}((b,+\infty ))\in \mathcal {B}_{\mathbb {X}}\). First, let us suppose that \(E\) is an open set on \(\mathbb {X}\times I_0\). Let \(\theta _0\) be any point in \(S^{-1}((b,+\infty ))\). Then there exists \(y_0\in I_0\) such that \(y_0 >b\) and \((\theta _0,y_0)\in E\). The openness of \(E\) shows the existence of open sets \(A\subset \mathbb {X}\) and \(B\subset I_0\) such that \((\theta _0,y_0)\in A\times B\subset E\). Thus \(A\subset S^{-1}((b,+\infty ))\) and it shows that \(S^{-1}((b,+\infty ))\) is an open set.

In the general case, given any measurable set \(E\), let us consider the sets

for \(n\in \mathbb {N}\). We consider the functions \(S_n(\theta )=\sup \{y\in I_0; (\theta ,y)\in B(E,1/n)\}\). These functions are measurable by what we have proved. Since \(S=\inf _{n\in \mathbb {N}} S_n\), the measurability of \(S\) follows.\(\square \)

Using the measurability of \(a_i(\theta ,x)\) and \(b_i(\theta ,x)\) we conclude the measurability of \(\theta \mapsto r_i(\theta ,x)\) (all for fixed \(x\in I_0\)). It finishes the proof of Lemma 9.3.

Now, we want to prove the measurability of the maps \(l^{*}_{j}\). Let us consider a sequence of measurable partitions \(\cdots \subset \mathcal {P}_{n+1}\subset \mathcal {P}_{n}\subset \cdots \subset \mathcal {P}_1\) of \(I_0\) such that the norm of \(\mathcal {P}_n\) is less than \(1/n\). Choose a point \(x^{n}_i\) in each \(P^{n}_i\) element of \(\mathcal {P}_n\) and define the functions

We also consider the map \(l_j\,{:=}\,\liminf _{n\rightarrow \infty } l^{n}_{j}\).

Lemma 9.5

The maps \(l_{j}:\mathbb {X}\times I_0\rightarrow \mathbb {R}\) are measurable for all \(j\in \mathbb {N}\).

Proof of Lemma 8.4

For fixed \(x\in I_0\), the maps \(\theta \rightarrow |f_{\theta }^{j}(T^j(\theta ,x))|\) are measurable, since

Therefore the maps \(l^{n}_{j}\) are measurable. Obviously it implies the measurability of maps \(l_j\).\(\square \)

Proof of Proposition 9.2

By Lemma 9.5, the map \(l_j\) is measurable and \(l_j(\theta ,x)=l^{*}_j(\theta ,x)\) if \(r_j(\theta ,x)>0\). By Lemma 9.3, the sets \(\mathcal {H}_i(\sigma )\,{:=}\,\{ z\in \mathbb {X}\times I_0; r_i(z)>\sigma \}\) are measurable, for any \(\sigma >0\). These facts imply that \(H_i(\sigma )=\mathcal {H}_i(\sigma )\cap {(l^{*}_{j})}^{-1}(3\sigma ,\infty )\) is a measurable set.

Let us fix a set \(J\in \mathcal {B}_{I_0}\). As on Proposition 9.1, to prove the measurability of \(\theta \mapsto \mu _n(\theta )(J)\) it suffices to prove the measurability of the functions \(\theta \mapsto \mu ^{i}_{J}(\theta ) {:=}\, (f_{\alpha ^{-i}(\theta )}^{i})_{*} (\mathrm{m }| H_{i}(\alpha ^{-i}(\theta ),\varsigma )\cap Z(\alpha ^{-i}(\theta ),\lambda ))(J)\), for \(i\in \mathbb {N}\). Now, we have that \(\mu ^{i}_{J}(\theta )= \int _{I_0} \phi _i(\theta ,x)\psi _i(\theta ,x)\, d\mathrm{m }(x)\), where \(\phi _i,\psi _i:\mathbb {X}\times I_0\rightarrow \mathbb {R}\), \(\phi _i\) are respectively defined in (9.1) and

Once again, using Fubini’s Theorem, the measurability of \((\theta ,x)\mapsto \phi _i(\theta ,x) \psi _i(\theta ,x)\) implies the measurability of \(\theta \mapsto \mu ^{i}_{J}(\theta )\).\(\square \)

1.3 Higher-dimensional fibers

1.3.1 The measures \(\eta _n\) are well defined

This case is precisely the same as the case with one-dimensional fibers, so we have nothing to add.

1.3.2 The measures \(\mu _n\) are well defined

From the definition of \(\mu _n\) in the higher dimensional case, we see that it is enough to show that for every \(n\in \mathbb {N}\) and Borel set \(S\subset \mathbb {Y}\) the function \(\mathbb {X}\ni \theta \mapsto \mu _n(\theta )(S)\) is measurable. For this it is enough to prove the following.

Lemma 9.6

The function \(\mathbb {X}\ni \theta \mapsto \mathrm{Leb }\big ( {\mathcal {H}}_j(\alpha ^{-j}(\theta )) \cap (f^{j}_{\alpha ^j(\theta )})^{-1}(S)\big )\) is measurable for each fixed \(j\in \mathbb {N}\) and measurable \(S\subset \mathbb {Y}\).

Analogously to the previous subsection, we consider the maps

and \(\chi _S\) the characteristic function of \(S\). These functions are all measurable with respect to the corresponding Borel \(\sigma \)-algebras. We consider also \(\chi _{{\mathcal {H}}_n}\) the characteristic function of \({\mathcal {H}}_n(\sigma ,\delta ,b)\).

Lemma 9.7

The set \({\mathcal {H}}_n(\sigma ,\delta ,b)\) is a Borel subset of \(\mathbb {X}\times \mathbb {Y}\).

Proof of Lemma 8.4

According to the definition of \((\sigma ,\delta ,b)\)-hyperbolic time

is an intersection of at most finitely many sets of the form \(\{(\theta ,x)\in \mathbb {X}\times \mathbb {Y}: g(\theta ,x)>c\}\) for a measurable function \(g:\mathbb {X}\times \mathbb {Y}\rightarrow \mathbb {R}\) and some constant \(c\in \mathbb {R}\). Indeed, if we define for \(k=0,\dots ,n-1\)

then we can write

Thus \({\mathcal {H}}_n(\sigma ,\delta ,b)\) is a Borel subset of \(\mathbb {X}\times \mathbb {Y}\) as soon as we show that \(g_k,d_k\) are measurable functions for each \(k\ge 0\).

Clearly \(g_k\) is measurable from condition \((H_6)\). For the functions \(d_k:\mathbb {X}\times \mathbb {Y}\rightarrow [0,+\infty )\) we clearly have

and we define

Clearly \(\xi :\mathbb {X}\times \mathbb {Y}\times \mathbb {Y}\rightarrow [0,\delta ]\) is measurable, so \(D:\mathbb {X}\times \mathbb {Y}\rightarrow [0,\delta ]\) is also measurable and \(d_k\) is a composition of \(D\) with other measurable maps from condition \((H_5)\). This completes the argument showing that \({\mathcal {H}}_n(\sigma ,\delta ,b)\) is a Borel subset of \(\mathbb {X}\times \mathbb {Y}\).\(\square \)

Now we are ready to prove the first lemma.

Proof of Proposition 9.6

We note that we can write

where

Since both \(\phi _j\) and \(\psi _j\) are Borel measurable from \(\mathbb {X}\times \mathbb {Y}\) to \(\mathbb {R}\), Fubini’s Theorem ensures that (9.2) is a measurable function of \(\theta \in \mathbb {X}\), as we need. This concludes the proof.\(\square \)

With Lemma 9.6 we complete the proof of the measurability of all functions used in the previous sections.

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