Recurrence and pressure for group extensions (Q2793101)

From the abstract: ``We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterizes recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincaré series of Kleinian groups and the cogrowth of group presentations.''NEWLINENEWLINELet \((\Sigma, \delta)\) be a countable Markov shift with countable alphabet \(I\). Let \(I^*\) denote the free semigroup generated by \(I\). Considering a countable group \(G\) and a semigroup homomorphism \(\Psi : I^* \to G\), a group-extended Markov system is defined as a skew product mapping points \((\omega, g) \in \Sigma \times G\) to \((\sigma, g\Psi(\omega_1))\), where \(\omega = (\omega_1, \omega_2,\ldots) \in \Sigma\).NEWLINENEWLINEThermodynamic formalism for countable Markov shifts was developed by \textit{O. M. Sarig} [Proc. Symp. Pure Math. 89, 81--117 (2015; Zbl 1375.37099)] and by \textit{R. D. Mauldin} and \textit{M. Urbański} [Graph directed Markov systems. Geometry and dynamics of limit sets. Cambridge: Cambridge University Press (2003; Zbl 1033.37025)]. Such a shift satisfying a very strong mixing condition is said to be ``finitely primitive'' or to satisfy Sarig's ``big images and preimages property.'' Such shifts have thermodynamical (and spectral) properties resembling those of full shifts and are well-behaved.NEWLINENEWLINEA potential or observable is a real-valued function on the space of interest. A potential may be recurrent (which is good) or positive recurrent (even better). In the latter case, the dynamical system has a finite equilibrium measure.NEWLINENEWLINEThe group \(G\) is amenable if it has a finitely-additive, left-invariant probability measure (there are numerous equivalent definitions).NEWLINENEWLINEThe author examines topologically-mixing extensions of finitely-primitive countable Markov shifts and potentials \(\phi : \Sigma \times G \to \mathbb{R}\) depending only on the base space \(\Sigma\) with finite Gurevič pressure. If the potential is recurrent, the group is shown to be amenable; if the potential is positive recurrent, the group must be finite. Properties of the equilibrium measure, pressure and Perron-Frobenius operator are given.