Critically Finite Random Maps of an Interval

TL;DR

This paper studies critically finite random multimodal maps of an interval, establishing existence and properties of conformal measures, pressure functions, and the Hausdorff dimension of the associated random Julia sets.

Contribution

It introduces the concept of critically finite random maps of an interval and proves key properties of conformal measures, pressure, and Hausdorff dimension in this setting.

Findings

Existence of t-conformal random measures for parameters in AA(T)

The expected topological pressure is independent of measure choice and is monotone decreasing

Hausdorff dimension of the random Julia set equals a critical parameter b_T

Abstract

We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $\Omega$ and a base invertible ergodic map $\theta:\Omega\to\Omega$ preserving a probability measure $m$ on $\Omega$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$-conformal random measure $\nu_t$ exists. We denote by $\lambda_{t,\nu_t,\omega}$ the corresponding generalized eigenvalues of the corresponding dual operators $\mathcal{L}_{t,\omega}^*$, $\omega\in\Omega$. (2) Given $t\ge 0$ any two $t$-conformal random measures are equivalent. (3) The expected topological pressure of the parameter $t$: $$\mathcal{E}P(t):=\int_{\Omega}\log\lambda_{t,\nu,\omega}dm(\omega) $$ is independent of the choice of a $t$-conformal random measure $\nu$. (4) The function $$ AA(T)\ni t\longmapsto \mathcal{E}P(t)\in\mathbb R $$ is monotone decreasing and Lipschitz continuous. (5) With $b_T$ being defined as the supremum of such parameters $t\in AA(T)$ that $\mathcal{E}P(t)\ge 0$, it holds that $$ \mathcal{E}P(b_T)=0 \ \ \ {\rm and} \ \ \ [0,b_T]\subset \text{Int}(AA(T)). $$ (6) $\text{HD}(\mathcal{J}_\omega(T))=b_T$ for $m$-a.e $\omega\in\Omega$, where $\mathcal{J}_\omega(T)$, $\omega\in\Omega$, form the random closed set generated by the skew product map $T$. (7) $b_T=1$ if and only if $\bigcup_{\Delta\in \mathcal{G}}\Delta=[0,1]$, and then $\mathcal{J}_\omega(T)=[0,1]$ for all $\omega\in\Omega$.