# Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in ${\mathbb{R}}^{d}$

Annales Polonici Mathematici (1998)

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

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topPiotr 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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