Invariant measures for random piecewise convex maps

For a given non-singular map on a probability space, the question whether an invariant measure, which is absolutely continuous with respect to the reference measure, exists or not is one of the fundamental problems in ergodic theory. The same question for a random map (in terms of both annealed and quenched sense) makes sense and is also important passing through ergodic theory for Markov operators, skew-product transformations or Markov operator cocycles. See [4, 13, 22, 21] and references therein. On the one hand, if a system, whether deterministic or random, admits an absolutely continuous finite invariant measure, some classical ergodic theorems such as the Birkhoff ergodic theorem are applicable. Moreover some limit theorems such as the central limit theorem may be expected [3, 10, 11]. On the other hand, if a system possesses only an absolutely continuous $\sigma$-finite and infinite invariant measure, such a system is within the scope of infinite ergodic theory and has been paid attention for recent decades [2, 25, 28]. Typical examples of such systems have indifferent or neutral, but weakly repelling, fixed points and are known as an intermittent model, as well as a model of non-uniformly hyperbolic systems. The existence of absolutely continuous $\sigma$-finite invariant measures and several statistical properties for random versions of intermittent maps thus have been recently enthusiastically studied [5, 6, 7, 18, 29, 30].

The subject of the present paper is a certain class of one-dimensional random dynamical systems called random piecewise convex maps in annealed (or i.i.d.) sense (see the conditions ( ‣ 2)–(2) and (A) or (B) precisely). The existence of invariant measures and their ergodic properties of the deterministic piecewise convex maps were firstly studied by Lasota and Yorke in [23] for the case when maps are uniformly expanding on the first branch and other branches have positive derivative, and then studied by Inoue in [15, 16] for more general cases. The aim of this paper is to generalize them, demonstrating that random piecewise convex maps admit Lebesgue-equivalent ergodic $\sigma$-finite invariant measures. (For some interesting studies of random generalization of piecewise convex maps with ``position dependent'' probability measures, we refer to [8] (cf., [20]), while we do not deal with position dependent random maps but we handle random maps consisting of potentially uncountably many maps with infinite invariant measures.) We also estimate the size of invariant measures, from which it is revealed whether the $\sigma$-finite invariant measures for random piecewise convex maps are finite or infinite. The phenomenon that an invariant measure varies from finite to infinite as a parameter of a system varies is well-known for (deterministic) intermittent maps employed by Thaler in [27] or Liverani–Saussol–Vaienti (see (1.1) below) in [24]. Although our random piecewise convex maps also admit both finite and $\sigma$-finite infinite invariant measures depending on parameters and probabilities of choice of maps, some examples of them (e.g., Example 4.1) have neither a common indifferent fixed point nor a common critical point, which is very different from deterministic cases. The mechanism is, roughly speaking, derived from strong contracting property on average, which never occurs for deterministic systems and is unique to random dynamical systems.

We then briefly review the LSV map, named after Liverani–Saussol–Vaienti from [24], which has been analyzed as a simple model of intermittency, and we will compare them (and known random versions of them, see also [5, 6, 7]) with our random piecewise convex maps. For $\alpha>0$, the LSV map $T_{\alpha}:[0,1]\to[0,1]$ is defined by

which has an indifferent fixed point 0 and Lebesgue-typical orbits would be trapped around a small neighborhood of 0 for a long time. For this map $T_{\alpha}$, it is well-known that a Lebesgue-equivalent ergodic invariant measure exists and that the invariant density function is of order $x^{-\alpha}$ as $x\to 0$ [24, 27, 32]. Thus $T_{\alpha}$ possesses an equivalent finite invariant measure for $0<\alpha<1$ and an equivalent $\sigma$-finite and infinite invariant measure for $\alpha\geq 1$. The order of invariant measures radically affects their statistical properties, such as the central limit theorem or the mixing rate in the finite measure-preserving case (cf., [7, 14, 26]) and the wandering rate in the Darling–Kac theorem or the arcsine law in the infinite measure-preserving case (cf., [2, 25, 28]). Therefore it is certainly worth establishing invariant measures and analyzing the asymptotics of the invariant measures for given systems. The asymptotics of the invariant density for $T_{\alpha}$ is also tightly related to the decay of the inverse images of the disconnected point $\frac{1}{2}$ by the left branch. If we set $x_{1}(\alpha)=\frac{1}{2}$ and $x_{n}(\alpha)\in[0,\frac{1}{2})$ for $n\geq 2$ such that $T_{\alpha}(x_{n+1}(\alpha))=x_{n}(\alpha)$ for $n\geq 1$, then it follows from the results in [32, 27] that

for some $C>0$, where $a_{n}\sim b_{n}$ for positive sequences $\{a_{n}\}_{n=1}^{\infty},\{b_{n}\}_{n=1}^{\infty}\subset\mathbb{R}$ stands for $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=1$. In this paper, for random piecewise convex maps, we establish the existence of Lebesgue-equivalent, conservative and ergodic $\sigma$-finite invariant measures and evaluate the asymptotics of the invariant measures, which is a generalization of results for random LSV maps as in [5, 6]. The advantage of our results is that we do not restrict ourselves to constituent maps of random dynamical systems to be at most countable nor expanding on average outside of a small neighborhood of the common indifferent fixed point. As an application, we can modify a random LSV map to admit uniformly contracting branches and moreover to admit a critical or flat point around the inverse image of an indifferent fixed point (see Example 4.3–4.6). The key point in the estimate of the invariant measures for random piecewise convex maps is, on the contrary to the LSV maps, the decay of random inverse images of the disconnected point by the right branches (see Definition 2.1, Theorem 3.2 and Theorem 3.3 precisely). That is, one needs to take the contracting effect by right branches into account. Indeed, the induced (random) map/the first return (random) map for (random) LSV maps satisfy uniformly expanding property (on average), whereas those for our random piecewise convex maps do not in general. Hence we cannot expect so-called spectral decomposition method from the Lasota–Yorke type inequality any longer. We refer to [30] for similar arguments and some background.

In this section, we define random picewise convex maps. Let $X=[0,1]$ be the unit interval equipped with the Lebsgue measure $\lambda$ over the Borel $\sigma$-algebra $\mathcal{B}$ and let $\mathbb{A}$ and $\mathbb{B}$ be some parameter regions with some measurable structure (usually they are subspaces of $\mathbb{N}$ or $\mathbb{R}$). For each $\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$, we define a non-singular maps $T_{\alpha,\beta}$, i.e., $\lambda(T_{\alpha,\beta}^{-1}N)=0$ whenever $\lambda(N)$=0, on $X$ by

where $\tau_{\alpha}:[0,\frac{1}{2}]\to X$ and $S_{\beta}:(\frac{1}{2},1]\to X$ for $\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$ are injective and continuous maps with some conditions (see the conditions ( ‣ 2)–(2), (A) and (B) precisely). The standing assumption on $\{\tau_{\alpha}:\alpha\in\mathbb{A}\}$ and $\{S_{\beta}:\beta\in\mathbb{B}\}$ is the following:

1. (0)

   The map $T:\mathbb{A}\times\mathbb{B}\times X\to X$; $(\alpha,\beta,x)\mapsto T_{\alpha,\beta}(x)$ is measurable with respect to each variable.

Note that the above condition ( ‣ 2) is fulfilled if $\mathbb{A}$ and $\mathbb{B}$ are topological spaces endowed with their Borel structures and the maps $\mathbb{A}\ni\alpha\mapsto\tau_{\alpha}$ and $\mathbb{B}\ni\beta\mapsto S_{\beta}$ are continuous.

Our random dynamical systems are defined as random compositions of maps $\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\}$ with the condition ( ‣ 2) in the annealed sense. In order to define our random dynamical systems, we set probability measures $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ on $\mathbb{A}$ and $\mathbb{B}$, respectively. $\nu_{\mathbb{A}}^{\infty}$ denotes the infinite product of the probability measure of $\nu_{\mathbb{A}}$ over $\mathbb{A}^{\mathbb{N}}$. Then, for the family of maps $\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\}$ and probability measures $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ over the parameter spaces $\mathbb{A}$ and $\mathbb{B}$, we consider the following transition function

for each $A\in\mathcal{B}$ and $\lambda$-almost every $x\in X$. By the condition ( ‣ 2) and non-singularity of each $T_{\alpha,\beta}$ with respect to $\lambda$, it is straightforward to see that this transition function is null-preserving, i.e., $\lambda(N)=0$ implies $\mathbb{P}(x,N)=0$ for $\lambda$-almost every $x\in X$. Thus, we can define the corresponding Markov operator $P:L^{1}(X,\lambda)\to L^{1}(X,\lambda)$ (i.e., $Pf\geq 0$ and $\lVert Pf\rVert_{L^{1}}=\lVert f\rVert_{L^{1}}$ for each $f\in L^{1}(X,\lambda)$ non-negative) by

for each $f\in L^{1}(X,\lambda)$ and $A\in\mathcal{B}$. The adjoint operator of $P$ acting on $L^{\infty}(X,\lambda)$ is denoted by $P^{*}$ which is characterized by

for each $f\in L^{1}(X,\lambda)$ and $g\in L^{\infty}(X,\lambda)$.

In order to make more precise assumptions on random piecewise convex maps, we introduce some notations. As in the previous section (note that $\tau_{\alpha}$ is not necessarily the same as the LSV map $T_{\alpha}|_{[0,\frac{1}{2}]}$ from §1), for $\bm{\alpha}=(\alpha_{1},\alpha_{2},\dots)\in\mathbb{A}^{\mathbb{N}}$, let $x_{1}^{\bm{\alpha}}=x_{1}\coloneqq\frac{1}{2}$ and $x_{n+1}^{\bm{\alpha}}\coloneqq\tau_{\alpha_{n-1}}^{-1}\circ\cdots\circ\tau_{\alpha_{1}}^{-1}(x_{1}^{\bm{\alpha}})$ for $n\geq 1$. For simplicity, let $x_{0}^{\bm{\alpha}}=x_{0}\coloneqq 1$ and set $X_{n}^{\bm{\alpha}}\coloneqq(x_{n+1}^{\bm{\alpha}},x_{n}^{\bm{\alpha}}]$ for $\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ and $n\geq 0$.

For considering the inverses by the right branch of $x_{n}^{\bm{\alpha}}$'s as well, we need the following definition:

We always assume that $\eta$ is measurable as a standing hypothesis. Then, for $\bm{\alpha}\in\mathbb{A}$ and $\beta\in\mathbb{B}$, let $y_{1}^{\bm{\alpha},\beta}=y_{1}\coloneqq 1$ and $y_{n+1}^{\bm{\alpha},\beta}$ be the inverse of $x_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}$ by the right branch of $T_{\alpha,\beta}$, namely,

We set $Y_{n}^{\bm{\alpha},\beta}\coloneqq(y_{n+1}^{\bm{\alpha},\beta},y_{n}^{\bm{\alpha},\beta}]$ for $n\geq 1$ and $Y\coloneqq[\frac{1}{2},1]$.

Throughout the paper we assume, together with the condition ( ‣ 2), that a family of maps $\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\}$ satisfies the following conditions (piecewise convex property, see also Figure 1): for $\nu_{\mathbb{A}}$-almost every $\alpha\in\mathbb{A}$ and $\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$,

1. (1)

   $\tau_{\alpha}$ and $S_{\beta}$ are $C^{1}$-functions and $S_{\beta}$ can be extended to a continuous function on $Y$ (the extension is also denoted by the same symbol $S_{\beta}$) with $\tau_{\alpha}(0)=0$, $\tau_{\alpha}(\frac{1}{2})=1$ and $S_{\beta}(\frac{1}{2})=0$;

2. (2)

   $\tau_{\alpha}^{\prime}$ and $S_{\beta}^{\prime}$ are non-decreasing on $(0,\frac{1}{2})$ and $(\frac{1}{2},1)$, respectively, with $\tau_{\alpha}^{\prime}(0)\geq 1$, $\tau_{\alpha}^{\prime}(x)>1$ for $x\in(0,\frac{1}{2})$, $S_{\beta}^{\prime}(\frac{1}{2})\geq 0$ and $S_{\beta}^{\prime}(x)>0$ for $x\in(\frac{1}{2},1)$, where $\tau_{\alpha}^{\prime}(0)$ and $S_{\beta}^{\prime}(\frac{1}{2})$ are taken as the right derivatives.

By our assumptions (1) and (2), for $\nu_{\mathbb{A}}^{\infty}$-almost every $\bm{\alpha}=(\alpha_{1},\alpha_{2},\dots)\in\mathbb{A}^{\mathbb{N}}$ and $\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$, we have $T_{\alpha_{n},\beta}X_{n}^{\bm{\alpha}}=X_{n-1}^{\bm{\alpha}}$ and $T_{\alpha,\beta}Y_{n+1}^{\bm{\alpha},\beta}=X_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}$ for any $n\geq 1$ and $T_{\alpha_{0},\beta}Y_{1}^{\bm{\alpha},\beta}=(x_{\eta(\bm{\alpha},\beta)+1}^{\bm{\alpha}},S_{\beta}(1)]\subset X_{\eta(\bm{\alpha},\beta)}^{\bm{\alpha}}$, where $\alpha_{0}\in\mathbb{A}$ is arbitrary.

Recall that for a Markov operator $P$, a measure $\mu$ over $(X,\mathcal{B})$ is called an absolutely continuous (resp. equivalent) $\sigma$-finite invariant measure if $\mu$ is a $\sigma$-finite measure which is absolutely continuous (resp. equivalent) with respect to $\lambda$ and its Radon–Nikodým derivative $\frac{d\mu}{d\lambda}$ is a (non-trivial) fixed point of $P$. Note that by positivity of Markov operators the domain of any Markov operator can be naturally extended to the set of non-negative and locally integrable functions and hence the definition of absolutely continuous $\sigma$-finite infinite invariant measures makes sense.

We then consider the following technical conditions on our random dynamical systems, which are important in establishing the existence of equivalent $\sigma$-finite invariant measures.

1. (A)

   $\displaystyle\operatorname*{ess\,sup}_{\alpha\in\mathbb{A}}\tau^{\prime}_{\alpha}\left(\tfrac{1}{2}\right)<\infty$;

2. (B)

   $\displaystyle\int_{\mathbb{A}}\frac{1}{x_{1}-x_{2}^{\bm{\alpha}}}d\nu_{\mathbb{A}}(\alpha)<\infty$.

Obviously the condition (A) implies the condition (B).

In what follows, $\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\}$ denotes a random dynamical system given by the transition function (2.1) with conditions ( ‣ 2)–(2) and is referred as a random piecewise convex map. In the next section, we prove the existence and uniqueness of equivalent $\sigma$-finite invariant measures for random piecewise convex maps. Furthermore, we show the asymptotics of the invariant measures.

Before stating the main results in the paper, some basic definitions are listed. Let $\mu$ be an absolutely continuous measure with respect to $\lambda$. Recall that an invariant set for a Markov operator $P$ is a measurable set $E\in\mathcal{B}$ with the property $P^{*}1_{E}=1_{E}$ $\lambda$-almost everywhere and $P$ is called ergodic with respect to $\mu$ if each invariant set $E$ satisfies either $\mu(E)=0$ or $\mu(X\setminus E)=0$. $P$ is called conservative with respect to $\mu$ if any function $h$ supported on $\operatorname*{supp}\mu$ with $h\geq P^{*}h$ satisfies $h=P^{*}h$. Other equivalent characterization of conservativity are found in [22]*§3.1.

The following main theorem establishes the existence of Lebesgue-equivalent, conservative and ergodic $\sigma$-finite invariant measures for random piecewise convex maps defined in the previous section.

We further state two theorems of Theorem 3.1 which tell us when the invariant measure becomes an infinite measure. The first one deals with a specific case when $\mathbb{A}$ is a point set, from which we can show the general case as in Theorem 3.3.

When $\mathbb{A}$ is an uncountable set, the form of the invariant density is complicated in general. However, combining Theorem 3.2 and the comparison theorem from [19], we can estimate the size of the $\sigma$-finite invariant measure $\mu$ in Theorem 3.1 even when $\mathbb{A}$ is not singleton, by reducing to the case of singleton. In order to clarify our statement, we need to introduce the following condition. A random piecewise convex map $\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\}$ is said to satisfy the condition ($\dagger$) if there are some $c\in(0,\frac{1}{2})$ and $\alpha_{1},\alpha_{2}\in\mathbb{A}$ such that

These conditions are of course equivalent to that

With some abuse of notation, for a fixed $\bar{\alpha}\in\mathbb{A}$, $\{T_{\bar{\alpha},\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\}$ denotes a random piecewise convex map $\{T_{\alpha,\beta};\nu_{\bar{\alpha}},\nu_{\mathbb{B}}:\alpha\in\{\bar{\alpha}\},\beta\in\mathbb{B}\}$ where $\nu_{\bar{\alpha}}$ is the Dirac measure on $\bar{\alpha}$.

Before proving Theorem 3.1, we recall the key tool, called the induced operator (or the first return map in the sense of [18]), to construct an absolutely continuous $\sigma$-finite invariant measure and we also prepare lemmas.

As in the previous section, we let $Y=[\frac{1}{2},1]$ and recall (see also [13, 29]) that the induced operator (on $Y$) $P_{Y}$ is defined by

where $I_{Y}$ and $I_{Y^{c}}$ are the restriction operators on $Y$ (i.e., $I_{Y}f=1_{Y}f$ for each measurable function $f$) and $Y^{c}$, respectively. The operator $P_{Y}$ is a well-defined Markov operator over $L^{1}(X,\lambda)$ since $Y$ is a $P$-sweep-out set with respect to $\lambda$ (see Lemma 4.7 in [29] precisely). The induced operator for a Markov operator is a generalization of the induced map for a non-singular map.

For $(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$, $\mathcal{L}_{T_{Y}^{(\bm{\alpha},\beta)}}$ denotes the Perron–Frobenius operator associated with the induced (random) map $T_{Y}^{(\bm{\alpha},\beta)}x\coloneqq\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\alpha_{n(x)}}\circ S_{\beta}x$ where $n(x)\geq 1$ is the minimum number satisfying $\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\alpha_{n(x)}}\circ S_{\beta}x\in Y$ (such $n(x)$ exists for $x\in Y\setminus\{\frac{1}{2}\}$).

We then prove the following key lemma.

We now emphasize that the left branches $\tau_{\alpha}$'s map points surjectively onto $[0,1]$. This together with the condition (B) guarantees that an invariant density for the induced operator $P_{Y}$ is fully supported on $Y$ as well as bounded above. Furthermore, (A) ensures the invariant density to be bounded away from zero on $Y$. Henceforth $\lambda|_{Y}$ denotes the measure $\lambda$ restricted on $Y$.