Unique equilibrium states for Bonatti–Viana diffeomorphisms

Abstract

We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under C 1 perturbations of the map. These results are an application of general machinery developed by the first and last named authors.

Denseness of intermediate pressures for systems with the Climenhaga-Thompson structures

“…Proof. By Lemma 3.7 and (8), for n > 10, we have For every m > 10N , there is n > 10 and l ∈ Z N such that m = (n − 3)N + l. So finally by (8) and (6) Then P e (X, f, φ) ⊃ min P * (φ), P ⊥ exp (φ, ε) , P (φ) . Proof.…”

“…For each n ∈ Z + , there are τ n−1 elements in Z n−1 τ . By (8) and the pigeonhole principle, there must be an element (v 1 , · · · , v n−1 ) ∈ (Z τ ) n−1 such that (y1,··· ,yn)∈E(v1,··· ,vn−1) ( n k=1 e Φ(y k ,N ) ) ≥ (y1,··· ,yn)∈E n ( n k=1 e Φ(y k ,N ) )…”

“…Recently, Climenhaga and Thompson developed a methodology to study uniqueness of equilibrium states for certain partially hyperbolic systems [5,6,7,8,9]. The core of their theory is a decomposition of orbit segments into pieces satisfying gluing orbit properties and pieces of low pressures.…”

Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map

“…The proof is based on the elementary trigonometry and basic cone estimate. For a detailed proof of a more general version, see Lemma 3.6 in [7].…”

“…The spirit is to generalize the dynamical properties for the map and regularity conditions for potential functions from [2] and make them hold on an "essential collection of orbit segments" which dominates in topological pressure and presents "enough uniformly hyperbolic behavior". This technique has been applied to other non-uniformly hyperbolic cases, see [7], [8] for DA (derived from Anosov) homeomorphisms, and [3] for flows. We will compare our approach to that of [20] after we state our results and explain the details in §7.…”