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Asymptotic properties of the sequences (a) {P j φg} ∞ j=1 and (b) {j ∑j−1 i=0 P i φg} ∞ j=1, where Pφ : L 1 → L is the Frobenius–Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = {f ∈ L : 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 L-convergence of the sequences (a) and (b) to a unique g0 ∈ G. The general result is applied to some smooth Markov maps in R. Also the Bernoulli property is proved for a class of smooth Markov maps in R.

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