On Discretizations of the Generalized Boole Type Transformations and their Ergodicity

Abstract

There is studied an analytical discretization of the generalized Boole type transformations in ℝn 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.

Entropy and Ergodicity of Boole-Type Transformations

“…owing to changing the variables x = cot(πs), y = cot(πt), (s, t) ∈ [0, 1) 2 , (x, y) ∈ R 2 , subject to the new coordinates (s, t) ∈ [0, 1) 2 and the transformation S −1 : [0, 1) 2 (s, t) → ({s + t}, {s − t}) ∈ [0, 1) 2 . That this approach could be used to prove of the ergodicity theorem of the two-dimensional Boole transformation (73), was announced in [70][71][72] and is now confirmed by the following result.…”

“…where a, b j ∈ R, j = 1, N, α, β j ∈ R + , j = 1, N, and analyzed in [1,3,11,72,73]. For α = 1, a = 0, the ergodicity result was proved in [3,[74][75][76] by making use of a specially devised inner function method.…”

“…for α = 1/2, and arbitrary a, b ∈ R and β ∈ R + . Invariant measures and ergodicity related to (67) were analyzed in [11,[70][71][72] using their equivalence to…”

“…In this regard, we should mention [14,61,62], which treat many interesting measure preserving and ergodic multi-dimensional mappings. Recently, in [11,70,72,73] a class of new multi-dimensional Boole type transformations β ρ : R n → R n of the following form were introduced and analyzed…”