On Conformal Measures and Harmonic Functions for Group Extensions

Abstract

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of σ-finite conformal measures and give applications to the construction of harmonic functions.

Thermodynamic Formalism for Transient Potential Functions

“…Later on, motivated by the study of KMS states, Thomsen had analyzed the class of conformal measures of Markovian potentials on countable-states Markov shifts, and produced examples where the dependence of this class on the potential exhibits a phase transition, see [39] and also [40]. Stadlbauer used the Patterson-Sullivan approach to construct eigenmeasures and eigenfunctions for a class of topological Markov shifts obtained from countable group extensions of topological Markov shifts with the big images and pre-images property, see [37]. For more on the Thermodynamic Formalism of a transient potential function, see [16].…”

Anosov flows, growth rates on covers and group extensions of subshifts

“…The proof is inspired by Roblin's proof in [35] that if is a convex cocompact group of isometries of a CAT(−1) space and is a normal subgroup such that / is amenable then the critical exponents of and are equal. We will make use of a family of σ -finite measures ν η,g , indexed by + × G, introduced by Stadlbauer [43]. For t > 0, write…”

“…We can also ensure that there is a countable dense subset of η for which the limit is attained along the same subsequence. In Theorem 5.1 of [43], it is shown how to extend this to all η ∈ + using Hölder continuity. For the Proof of our Theorem 5.1, we will only use the countable collection of points…”

The conformal measures of a normal subgroup of a cocompact Fuchsian group

“…In §3 we show that for every δ ≥ δ Ŵ there is a linear one-to-one correspondence between the extremal (Ŵ, δ)-conformal measures and eigenmeasures of a suitable Ruelle operator. As the theory of the eigenmeasures of the Ruelle operator is well developed (see [10,18,35,41,42,49,51]), this correspondence can be a step towards a classification of the conformal measures. In particular, in [49] the author presented the eigenmeasures of a transient Ruelle operator (see the definitions in §2.2) in terms of points on a Martin boundary.…”