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Recurrence criterion for homogeneous spaces. (Un critère de récurrence pour certains espaces homogènes.) (French. English summary) Zbl 1417.37077

Let \(G\) be a real connected algebraic semi-simple Lie group, and \(H\) an algebraic subgroup of \(G\). Let \(\mu\) be a probability measure on \(G\), with finite exponential moment, and whose support spans a Zariski-dense subsemigroup of \(G\). Let \(X = G/H\) be the quotient of \(G\) by \(H\). The author is interested in the Markov chain on \(X\) with transition probability \(P_x = \mu * \delta_x\), for \(x \in X\). She proves that, either for every \(x \in X\) almost every trajectory starting from \(x\) is transient, or for every \(x \in X\) almost every trajectory starting from \(x\) is recurrent. The recurrence is uniform over all \(X\), i.e., there exists a compact set \(C \subset X\) such that for each point \(x \in X\), every trajectory starting in \(x\) almost surely returns to \(C\) infinitely often. Furthermore, she gives a criterion for recurrence depending on \(G, H\), and \(\mu\).

MSC:

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
22E46 Semisimple Lie groups and their representations
22D40 Ergodic theory on groups
Full Text: DOI

References:

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