Thermodynamic formalism on generalized countable Markov shifts

Abstract: Given a 0-1 infinite matrix $A$, R. Exel and M. Laca have introduced a kind of \textit{generalized countable Markov shift} (GCMS) $X_A=\Sigma_A \cup Y_A$, which is a locally compact (in many important cases compact) version of $\Sigma_A$, the standard countable Markov shift. The elements of $Y_A$ are finite words, possibly including multiplicities. We develop the thermodynamic formalism for GCMS, where we introduced the notion of conformal measure on $X_A$, and we explored its connections with the usual formalism on $\Sigma_A$. Among the results, we highlight the finding of new conformal measures that are not detected by the thermodynamic formalism on $\Sigma_A$ and new phase transition phenomena: for a wide class of GCMS and potentials, we determined regions for the inverse of the temperature $\beta$, where we absence\existence of these new conformal probabilities, living on $Y_A$. The Gurevich entropy $h_G$ plays a fundamental role in determining these regions since the critical value for gauge potentials is $h_G$ when finite. We also have phase transition results for $h_G = \infty$, including the full shift. In addition, for the eigenmeasures of Ruelle's transformation, we discovered a length-type phase transition in the renewal shift: the existence of a critical value for $\beta$ where the measure passes from living on $\Sigma_A$ to live on $Y_A$. We showed that the notion of pressure introduced by M. Denker and M. Yuri for Iterated Function Systems (IFS) is a natural definition of pressure for $X_A$, and it coincides with the Gurevich pressure for GCMS basically for the same generality on which the thermodynamic formalism is developed for the standard countable Markov shifts and potentials.

Joint work with R. Bissacot (University of São Paulo (USP), Brazil), R. Exel (Federal University of Santa Catarina (UFSC), Brazil), and R. Frausino (University of Wollongong (UOW), Australia).

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University