It is proved that a doubly stochastic operator P is weakly asymptotically cyclic if it almost overlaps supports. If moreover P is Frobenius-Perron or Harris then it is strongly asymptotically cyclic.
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Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence $g_n ∈ G$ such that $μ^{∗n}(g_n A) ≡ 1$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power $μ^{∗k}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological group G.
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Let (X,d) be a metric space where all closed balls are compact, with a fixed σ-finite Borel measure μ. Assume further that X is endowed with a linear order ⪯. Given a Markov (regular) operator P: L¹(μ) → L¹(μ) we discuss the asymptotic behaviour of the iterates Pⁿ. The paper deals with operators P which are Feller and such that the μ-absolutely continuous parts of the transition probabilities ${P(x,·)}_{x∈X}$ are continuous with respect to x. Under some concentration assumptions on the asymptotic transition probabilities $P^{m}(y,·)$, which also satisfy inf(supp Pf₁) ⪯ inf(supp Pf₂) whenever inf(supp f₁) ⪯ inf(supp f₂), we prove that the iterates Pⁿ converge in the weak* operator topology.
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We show that the set of those Markov semigroups on the Schatten class 𝓒₁ such that in the strong operator topology $lim_{t→ ∞}T(t) = Q$, where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.