Random non-hyperbolic exponential maps

TL;DR

This paper studies the dynamics of random iterations of exponential maps, establishing existence and uniqueness of conformal and invariant measures, and relating the Hausdorff dimension of Julia sets to a zero of a pressure function.

Contribution

It introduces a framework for analyzing random exponential maps on the cylinder, proving measure existence, uniqueness, and linking Hausdorff dimension to a topological pressure zero.

Findings

Existence of unique random conformal measures supported on radial Julia sets

Existence of unique invariant measures absolutely continuous w.r.t. conformal measures

Hausdorff dimension of Julia sets equals the zero of an expected topological pressure

Abstract

We consider random iteration of exponential entire functions, i.e. of the form ${\mathbb C}\ni z\mapsto f_\lambda(z):=\lambda e^z\in\mathbb C$, $\lambda\in{\mathbb C}\setminus \{0\}$. Assuming that $\lambda$ is in a bounded closed interval $[A,B]$ with $A>1/e$, we deal with random iteration of the maps $f_\lambda$ governed by an invertible measurable map $\theta:\Omega\to\Omega$ preserving a probability ergodic measure $m$ on $\Omega$, where $\Omega$ is a measurable space. The link from $\Omega$ to exponential maps is then given by an arbitrary measurable function $\eta:\Omega\longmapsto [A,B]$. We in fact work on the cylinder space $Q:={\mathbb C}/\sim$, where $\sim$ is the natural equivalence relation: $z\sim w$ if and only if $w-z$ is an integral multiple of $2\pi i$. We prove that then for every $t>1$ there exists a unique random conformal measure $\nu^{(t)}$ for the random conformal dynamical system on $Q$. We further prove that this measure is supported on the, appropriately defined, radial Julia set. Next, we show that there exists a unique random probability invariant measure $\mu^{(t)}$ absolutely continuous with respect to $\mu^{(t)}$. In fact $\mu^{(t)}$ is equivalent with $\nu^{(t)}$. Then we turn to geometry. We define an expected topological pressure $\mathcal E P(t)\in{\mathbb R}$ and show that its only zero $h$ coincides with the Hausdorff dimension of $m$--almost every fiber radial Julia set $J_r(\omega)\subset Q$, $\omega\in\Omega$. We show that $h\in (1,2)$ and that the omega--limit set of Lebesgue almost every point in $Q$ is contained in the real line $\mathbb R$. Finally, we entirely transfer our results to the original random dynamical system on $\mathbb C$. As our preliminary result, we show that all fiber Julia sets coincide with the entire complex plane $\mathbb C$.