Thermodynamics of towers of hyperbolic type

Abstract

We introduce a class of continuous maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a compact topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> possesses a unique equilibrium measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript phi"> <mml:semantics> <mml:msub> <mml:mi> μ </mml:mi> <mml:mi> φ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , associated to each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the Central Limit Theorem. We then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable. Our results extend those obtained in previous works of the first and second authors for inducing schemes of expanding types and apply to certain multidimensional maps. Applications include obtaining the thermodynamic formalism for Young’s diffeomorphisms, the Hénon family at the first bifurcation and the Katok map. In particular, we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t 0 greater-than t greater-than 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t_0&gt; t&gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t 0 greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t_0&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .