Existence of ACIM for Piecewise Expanding $C^{1+\varepsilon}$ maps

TL;DR

This paper proves the existence of an absolutely continuous invariant measure for certain piecewise expanding maps on an interval, using inequalities and ergodic theorems, and explores their mixing and decay properties.

Contribution

It establishes the existence of ACIM for $C^{1+ ext{epsilon}}$ maps via Lasota-Yorke inequalities and ergodic theory, extending previous results to this class of maps.

Findings

Existence of ACIM for piecewise expanding $C^{1+ ext{epsilon}}$ maps.

Proven quasi-compactness of the Frobenius-Perron operator.

Demonstrated weak mixing and exponential decay of correlations.

Abstract

In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+\varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem \cite{ionescu1950}, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius-Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.