Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

TL;DR

This paper proves the quasi-compactness of the Frobenius-Perron operator for piecewise convex maps with countably many branches, establishing ergodic properties and existence of invariant measures using advanced inequalities and ergodic theorems.

Contribution

It demonstrates the quasi-compactness of the Frobenius-Perron operator for complex maps with infinitely many branches, extending ergodic theory results.

Findings

Existence of absolutely continuous invariant measure (ACIM)

Quasi-compactness of Frobenius-Perron operator

Strong ergodic properties of the system

Abstract

In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $\tau$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $\tau$, $\tau^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $\mu$ for $\tau$, the exactness of the dynamical system $(I, \tau,\mu)$ and the quasi-compactness of Frobenius-Perron operator $P_\tau$ induced by $\tau$. The last fact implies a multitude of strong ergodic properties of $\tau$.