Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations

Abstract

Abstract. Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius-Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.

On Discretizations of the Generalized Boole Type Transformations and their Ergodicity

“…defined for any (x 1 , x 2, x 3 )∈ ℝ 3 \ {(0,0,0)} It was already proved in ref. [16] that these mapping are invariant with respect to the standard Lebesgue measure dv(x 1 , x 2, x 3 )=dx 1 dx 2 dx 3 on ℝ 3 yet their ergodicity is still under investigation.…”

“…), = 0, 1, exp ik q k q π − of the matrix  ϕ,N; x∈R Thus; the invariant infinitesimal measure with respect to the Boole mapping (12) equals subject to the standard Lebesgue measure dx on R Thus; one can formulate the following theorem. Being unique; modulo the constant multiplier; the invariant with respect to the Boole mapping (12) measure expression(16) is ergodic on axis ℝHaving now constructed the uniformly discretized Frobenius-Perron operator matrix (7); one can check that the matrix  ϕ,N is reducible with respect to any partition  N Then; based on Proposition 2.2; one can claim that the Boole mapping (12) is ergodic with respect to any partition  N ,N→∞ One can also verify that the positive definite vector (0) = (1 / ,1 / ,...,1 / ) N H N N N ∈  solves the limiting condition (8); being its eigenvector for the unity eigenvalue:…”

Remark on Transitivity for piecewise increassing maps on R

“…x es ergódica respecto a la medida de Lebesgue. Esto motivó una serie de trabajos relacionados con teoría ergódica infinita, ver [1,2,8,9,12] y recientemente [4,5,6,13,14]. Algunos de estos trabajos inclusive consideran parametrizaciones de la transformación de Boole de la siguiente forma:…”