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

TL;DR

This work proves a Lasota–Yorke inequality for the Frobenius-Perron operator of piecewise expanding $C^{1+\varepsilon}$ interval maps on the BV spaces $BV_{1,1/p}$, with $\varepsilon=1/p$, and uses the Ionescu–Tulcea–Marinescu theorem to obtain quasi-compactness and a finite spectral decomposition. It extends Keller’s generalized BV framework to $BV_{t,1/p}$, establishing uniform iterate bounds and the existence of absolutely continuous invariant measures on ergodic components, along with exactness properties for some iterates. Under a unique ACIM, the paper proves exponential decay of correlations both in the standard and invariant-measure formulations, reflecting strong mixing behavior. These results advance understanding of the ergodic and statistical properties of piecewise expanding $C^{1+\varepsilon}$ interval maps and provide a rigorous foundation for their spectral and mixing structure.

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.

Figures (2)

• Figure 1: The sets $E_i^L, F_i^L, E_i^R$ and $F_i^R$.
• Figure :

Theorems & Definitions (49)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• proof
• Theorem 2.7
• proof