Induced topological pressure for countable state Markov shifts

TL;DR

This paper introduces a new concept of induced topological pressure for countable state Markov shifts, linking thermodynamical quantities, entropy, and group amenability through a generalized framework.

Contribution

It generalizes existing notions of entropy and pressure for Markov shifts, incorporating a scaling function and subset of words, and connects these to group amenability and thermodynamical properties.

Findings

New definition of induced topological pressure for countable Markov shifts.

Established equivalence with classical pressure for certain functions.

Connected pressure concepts to group amenability and exhaustion principles.

Abstract

We introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words. Firstly, the scaling function allows a direct access to important thermodynamical quantities, which are usually given only implicitly by certain identities involving the classically defined pressure. In this context we generalise Savchenko's definition of entropy for special flows to a corresponding notion of topological pressure and show that this new notion coincides with the induced pressure for a large class of H\"older continuous height functions not necessarily bounded away from zero. Secondly, the dependence on the subset of words gives rise to interesting new results connecting the Gurevi{\vc} and the classical pressure with exhausting principles for a large class of Markov shifts. In this context we consider dynamical group extentions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.