Smoothness of invariant density for expanding transformations in higher dimensions

Kourosh Adl-Zarabi,Harald Proppe

doi:10.1155/s1048953300000046

Kourosh Adl-Zarabi, Harald Proppe

Open Access

https://doi.org/10.1155/s1048953300000046

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Abstract

Let Ω be a bounded region in ℝn and let 𝒫={Pi}i=1m be a partition of Ω into a finite number of closed subsets having piecewise C2 boundaries of finite (n−1)-dimensional measure. Let τ:Ω→Ω be an expanding transformation on 𝒫 where, τi:τ|Pi, and τi∈CM, m≥2. We show that the τ-invariant density h∈CM−2.

Highlights

Introduction-I I There has been a recent surge of interest in the study of existence and properties of absolutely continuous invariant measures (acim) of higher dimensional transformations
Dawson College, Department of Mathematics 300 Sherbrooke St
-I I There has been a recent surge of interest in the study of existence and properties of absolutely continuous invariant measures of higher dimensional transformations

Summary

Introduction

-I I There has been a recent surge of interest in the study of existence and properties of absolutely continuous invariant measures (acim) of higher dimensional transformations. Szewc [22] proved that the densities of invariant measures for Lasota-Yorke maps of class CM are of class CM- 1 It should be noted, that these smoothness properties are assumed to hold only piecewise that is, relative to a partition of The smoothness of the invariant density in Szewc’s result is piecewise-smoothness relative to another partition that is obtained from the given one through refinement with all of its forward images. M, tion is expanding in the sense of the maps considered by Man [16] and of class C its invariant density is of class CM- 2 It should be noted, that this somewhat weaker smoothness result is valid on the original partition. Some applications of expanding maps in a number theoretical context can be found in [20]

Definitions and Conditions

Main Results

Proofs and Laminas

Conclusions