Gurevic pressure and equidistribution for amenable extensions of countable state Markov shifts
Richard Sharp
TL;DR
The paper studies equidistribution of weighted periodic points for amenable skew-product extensions of mixing countable-state Markov shifts with BIP, establishing a weighted convergence to a Gibbs measure $\mu_{\varphi+\langle \xi,\bar{\psi}\rangle}$ under a moment condition. It proves a fundamental equality of Gurevič pressures $P_{G}(\varphi,T_\psi)=P_{G}(\varphi,T_{\bar{\psi}})$ when the extension group is amenable, and develops a framework (Assumptions I–III) in which $\xi$ minimizes the pressure and governs the equidistribution via large deviations arguments. The main contributions include (i) a generalization of equidistribution results to countable-state shifts with amenable skew products, (ii) a robust equality of Gurevič pressures enabling the abelianized analysis, and (iii) a converse characterization showing amenability from pressure equality under BIP. These results extend finite-type and flow-based results to broader symbolic dynamics, providing a versatile approach for dynamical systems with amenable covers and potential applications in thermodynamic formalism and geometric group theory.
Abstract
We obtain a weighted equidistribution theorem for amenable skew product extensions of countable state Markov shifts satisfying the BIP property. We also show, without requiring the BIP property, that Gurevic pressure for an amenable skew product agrees with the Gurevic pressure for the abelianized system. This had been proved by Dougall and Sharp in the case where the base is a subshift of finite type. The equality of Gurevič pressures is a key part of the proof of the equidistribution result.