Uniqueness of equilibrium measure for a family of partially hyperbolic horseshoes

In this paper, we show the uniqueness of equilibrium state for a family of partially hyperbolic horseshoes, introduced in Díaz et al. [Destroying horseshoes via heterodimensional cycles: Generating bifurcatios inside homoclinic classes, Ergod. Theory Dyn. Syst. 29 (2009), pp. 433–474] for some classes of continuous potentials. For the first class, the method used here is making use of the Sarig's theory for countable shifts. For this purpose, we study the dynamics of an induced map associated to the horseshoe map, we build a symbolic system with infinitely many symbols that is topologically conjugated to this induced map and we show that the induced potential is locally Hölder and recurrent. For the second class, we prove that uniqueness for the horseshoe is equivalent to uniqueness for the restriction to a non-uniformly expanding map, which is a hyperbolic potential and then it has uniqueness. Both classes include potentials with high variation, differently from previous results for potentials with low variation. We also prove uniqueness when the potential presents its supremum at a special fixed point and less than the pressure.