Homoclinic processes and invariant measures for hyperbolic toral automorphisms

M. I. Gordin

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 15, Tome 368 (2009), pp. 122-129

Résumé

For every hyperbolic toral automorphism $T$, the present author has defined in one of his previous papers some unbounded $T$-invariant second order difference operators related to the so-called homoclinic group of $T$. These operators were considered in the space $L_2$ with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism $T$. After a short discussion, some open questions about properties of these measures and related topics are posed. Bibl. – 9 titles.