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

Piotr Bugiel¹

¹

Annales Polonici Mathematici, Tome 68 (1998) no. 2, pp. 125-157

Résumé

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 $ℝ^d$.

Détail

DOI : 10.4064/ap-68-2-125-157

Keywords: invariant measure, Frobenius-Perron operator, expanding map, distortion inequality