Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in ℝ d

Piotr Bugiel

Annales Polonici Mathematici (1998)

  • Volume: 68, Issue: 2, page 125-157
  • ISSN: 0066-2216

Abstract

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Asymptotic properties of the sequences (a) P φ j g j = 1 ∞ and (b) j - 1 ∑ i = 0 j - 1 P ⁱ φ g j = 1 ∞ , where P φ : L ¹ → L ¹ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov maps in ℝ d . Also the Bernoulli property is proved for a class of smooth Markov maps in ℝ d .

How to cite

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Piotr Bugiel. "Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^d$." Annales Polonici Mathematici 68.2 (1998): 125-157. <http://eudml.org/doc/270740>.

@article{PiotrBugiel1998,
abstract = {Asymptotic properties of the sequences (a) $\{P^j_φ g\}_\{j=1\}^\{∞\}$ and (b) $\{j^\{-1\} ∑_\{i=0\}^\{j-1\} Pⁱ_φ g\}_\{j=1\}^\{∞\}$, where $P_φ:L¹ → L¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov maps in $ℝ^d$. Also the Bernoulli property is proved for a class of smooth Markov maps in $ℝ^d$.},
author = {Piotr Bugiel},
journal = {Annales Polonici Mathematici},
keywords = {invariant measure; Frobenius-Perron operator; expanding map; distortion inequality; Perron-Frobenius operator; Markov maps; Rényi condition; ergodic averages; recurrence; Bernoullicity},
language = {eng},
number = {2},
pages = {125-157},
title = {Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^d$},
url = {http://eudml.org/doc/270740},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Piotr Bugiel
TI - Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^d$
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 2
SP - 125
EP - 157
AB - Asymptotic properties of the sequences (a) ${P^j_φ g}_{j=1}^{∞}$ and (b) ${j^{-1} ∑_{i=0}^{j-1} Pⁱ_φ g}_{j=1}^{∞}$, where $P_φ:L¹ → L¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov maps in $ℝ^d$. Also the Bernoulli property is proved for a class of smooth Markov maps in $ℝ^d$.
LA - eng
KW - invariant measure; Frobenius-Perron operator; expanding map; distortion inequality; Perron-Frobenius operator; Markov maps; Rényi condition; ergodic averages; recurrence; Bernoullicity
UR - http://eudml.org/doc/270740
ER -

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