Beyond Bowen's Specification Property

Abstract

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen's arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.

Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved Climenhaga–Thompson criterion*

“…By the Bowen property on G 1 at scale ε > 6ρ, we obtain ( f t p (y), ρ), and remark that it is a disjoint union due to f t p (E t ) being (T(t), 2ρ)-separated11 . By the lower Gibbs property on G 1 in lemma 7.1 at scale ρ, it follows thatμ(V t ) =On the other hand, since E t ⊂ E t ∩ U t , we have Since U t is a union of elements of the partition A t which is adapted to the (t, 4ρ)-separated set E t , we see that for every y ∈ E t ∩ U t , there is a y ∈ Q t such that d t (y, y ) < 4ρ; this further implies that d T(t) ( f t p (y), f t p (y )) < 4ρ.…”

“…(II) The potential function φ has bounded distortion (the Bowen property) on G at a given scale ε > 40δ; 4 (III) The 'bad' parts of the system, consisting of P, S, D c and those points where the system is not expansive, must have smaller pressure comparing to G. Under these assumptions, they prove that there exists a unique equilibrium state with the upper and lower Gibbs property. For an overview of their result and applications, see the recent survey [11] and the references therein.…”

Existence and uniqueness of equilibrium states for systems with specification at a fixed scale

“…Under these assumptions, they prove that there exists a unique equilibrium state with the upper and lower Gibbs property. For an overview of their result and applications, see the recent survey [11] and the references therein.…”