Sensitive Dependence of Geometric Gibbs States at Positive Temperature

Abstract

We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature. * In §3.1 we extend the definition of essentially topologically exact map to certain quadratic-like maps that are not necessarily real.On the other hand, for t in R/Z the setis called the external ray of angle t of M. We say that R(t) lands at a point z in C, if Φ −1 (r exp(2πit)) converges to z as r ց 1. When this happens z belongs to ∂M.Note that for a real parameter c, every puzzle piece intersecting the real line is invariant under complex conjugation. Since puzzle pieces are simply-connected, it follows that the intersection of such a puzzle piece with R is an interval.Definition 2.2 (Yoccoz para-puzzle † ). Given an integer n ≥ 0, put J n := {t ∈ [1/3, 2/3] | 2 n t (mod 1) ∈ {1/3, 2/3}}, let X n be the intersection of W with the open region in the parameter plane bounded by the equipotential E(2 −n ) of M, and put I n := ∂X n ∪ X n ∩ t∈Jn cl(R(t)) .Then the Yoccoz para-puzzle of W is the sequence of graphs (I n ) +∞ n=0 . The parapuzzle pieces of depth n are the connected components of X n \ I n . The para-puzzle piece of depth n containing a parameter c is denoted by P n (c). † In contrast to [Roe00], we only consider the para-puzzle in the wake W.

Ergodic optimization in dynamical systems

“…The function H is concave, and extends continuously to the boundary of [α(f ), β(f )], though the absence of equilibrium measures m tf with f dm tf on the boundary prompted investigation of extremal measures (see [48,80,82,139]), and of the (typical) values H(α(f )) and H(β(f )) (see [145]). Finally, we note that zero temperature limits of equilibrium measures have been studied in a variety of other dynamical settings, including Frenkel-Kontorova models [6], quadratic-like holomorphic maps [54], multimodal interval maps [78], and Hénon-like maps [151].…”

Phase transitions for surface diffeomorphisms

“…In this broad terminology, phase transitions have been shown for several maps beyond the context of uniform hyperbolicity (see e.g. [8,9,10,11,16,20,28] just to mention a few). Following [2], we say that f has a phase transition with respect to the potential φ : X → R if the topological pressure function Bomfim and Carneiro asked whether it is possible to describe the mechanisms responsible for the existence of phase transitions for C 1 -local diffeomorphisms with positive topological entropy and Hölder continuous potentials, and obtained a fine description of the pressure function for local diffeomorphisms on the circle.…”