On Discretizations of the Generalized Boole Type Transformations and their Ergodicity

Prykarpatski Ak

doi:10.4172/2090-0902.1000199

Abstract

There is studied an analytical discretization of the generalized Boole type transformations in ân and their ergodicity properties.The fixed points of the corresponding finite-dimensional stochastic Frobenius-Perron operator discretization are constructed, the structure of the related invariant measures is analyzed

Highlights

The Frobenius-Perron Operator and Its DiscretizationWe consider an m-dimensional; not necessary compact; C1manifold Mm, endowed with a Lebesgue measure μ determined on the σ-algebra of Borel subsets of M m andφ: M m→M m being an almost everywhere smooth mapping
The described approach to study the dynamical properties of the mapping φ:Mm→Mm by means of the discretized Frobenius-Perron operator (6) is widely used in the literature [6,8,9,10,11]
If to state that this invariant measure is unique on the plane R2 this will mean [2,3,4,5] that the mapping φ1:R2→R2 is ergodic

Summary

Introduction

We consider an m-dimensional; not necessary compact; C1manifold Mm, endowed with a Lebesgue measure μ determined on the σ-algebra of Borel subsets of M m andφ: M m→M m being an almost everywhere smooth mapping. As a consequence of the definitions above one obtains that the discretized Frobenius-Perron operator (6) can be represented with respect to the canonical basis in the finite-dimensional space N by means of the (N×N) matrix φ,N = { φi,jN := μ(φ −1(Bi ) ∩ Bj )μ(Bj )−1 : i, j = 1, N},. The described approach to study the dynamical properties of the mapping φ:Mm→Mm by means of the discretized Frobenius-Perron operator (6) is widely used in the literature [6,8,9,10,11]. Namely; the ergodicity of it with respect to the partition N is defined as the irreducibility of the discretized Frobenius-Perron operator (6); and the mixing with respect to the partition N is defined as its primitivity and ergodicity

Discrete Ergodicity Analysis

The Classical Boole Mapping and Its Ergodicity

The generalized Boole Type Mapping and Its Ergodicity