In 1999, R. Exel and M. Laca solved the problem of extending the Cuntz-Krieger algebras to infinite alphabets. From this construction, the notion of generalized countable Markov shift (GCMS) arises, a completion of the usual Markov shift spaces (CMS) and depends only on the transition matrix, by including families of finite words that are invariant under the shift action. This new space is always locally compact, and even for a large class of non-locally compact CMS', their corresponding GCMS' are compact.

We developed the thermodynamic formalism for this generalized context by extending notions of eigenmeasures of the Ruelle's transformation, conformal measures, Gurevich pressure, etc.

We proved that the pressure of a point, a notion for pressure that considers the finite words constructed by M. Denker and M. Yuri, coincides with the Gurevich pressure for a wide class of potentials and GCMS. New conformal and eigenmeasures were discovered, as well as new phase transition phenomena.

In particular we emphasize the length-type phase transition, where the eigenmeasure passes from living on the CMS to its complement when we cool down the system.

A complete topological description of the GCMS was developed and allowed us to connect the new eigenmeasures and conformal measures via weak$^*$-limits on the inverse of temperature parameter.

This is a joint work with R. Bissacot, R. Exel and R. Frausino.