Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges

TL;DR

This paper investigates the dynamics of randomly composed Liverani-Saussol-Vaienti maps over large parameter ranges, revealing conditions under which the system exhibits absolutely continuous measures and decay of correlations.

Contribution

It extends previous work by analyzing compositions with parameters beyond [0,1], showing how mixed behaviors influence invariant measures and correlation decay.

Findings

Stationary measure is absolutely continuous if parameters less than 1 are sampled.

Decay rates of correlations match the fastest among sampled systems.

Rescaled Birkhoff averages converge to limit laws.

Abstract

Liverani-Saussol-Vaienti (L-S-V) maps form a family of piecewise differentiable dynamical systems on $[0,1]$ depending on one parameter $\omega\in\mathbb R^+$. These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations ($\omega <1$), or a unique physical measure that is singular and concentrated at the neutral point ($\omega >1$). In this paper, we study the composition of L-S-V maps whose parameters are randomly sampled from a range in $\mathbb R^+$, and where these two contrasting behaviours are mixed. We show that if the parameters $\omega<1$ are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where $\omega \in [0,1]$, we allow $ \omega >1$ in our sampling distribution. We also show that one can obtain similar decay of correlation rates for $\omega \in [0,\infty)$, when sampling is done with respect to a family of smooth, heavy-tailed distributions.