A self-consistent dynamical system with multiple absolutely continuous invariant measures

TL;DR

This paper investigates a class of self-consistent dynamical systems, revealing how the number of invariant measures changes with parameters, demonstrating phase transition-like behavior, and analyzing stability through numerical simulations.

Contribution

It introduces a new class of self-consistent systems with parameter-dependent invariant measures, including phase transition phenomena, and provides stability analysis via numerical methods.

Findings

Unique acim at zero self-consistency parameter

Multiple acims for positive self-consistency parameter

Phase transition-like behavior with varying parameters

Abstract

In this paper we study a class of \emph{self-consistent dynamical systems}, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of M. Blank, we concentrate on a special case where the dynamics in each step is a $\beta$-map with some $\beta \geq 2$. Included in the definition of $\beta$ is a parameter $\varepsilon > 0$ controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for $\varepsilon=0$, but at least two for any $\varepsilon > 0$. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for $0< \varepsilon < \varepsilon^*$, and multiple for sufficiently large values of $\varepsilon$. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.