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

Asymptotic properties of the sequences (a) ${\displaystyle {\{P^j\_\phi g\}}_{j=1}^{\infty }}$ and (b) ${\displaystyle {\left\{j^{-1}{\sum }_{i=0}^{j-1}P{ⁱ}_{\phi }g\right\}}_{j=1}^{\infty }}$, where ${\displaystyle P_{\phi }:L¹\to L¹}$ 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 ${\displaystyle {ℝ}^d}$. Also the Bernoulli property is proved for a class of smooth Markov maps in ${\displaystyle {ℝ}^d}$.