An extension of a theorem of Kesten to topological Markov chains

Details An extension of a theorem of Kesten to topological Markov chains An extension of a theorem of Kesten to topological Markov chains

2011-10-13

arxiv.org/abs/1110.2942v2

Mathematics

Dynamical Systems

The proof of theorem 5.4 is now replaced by a completely self-contained one. Furthermore, a short comment on Gibbs-Markov maps

Scientific paper

The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevic-pressures of the extension and the base coincide whereas the converse holds true if the potential is H\"older continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given.

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