Ground states and zero-temperature measures at the boundary of rotation sets

Abstract

We consider a continuous dynamical system f : X → X on a compact metric space X equipped with an m-dimensional continuous potential Φ = (φ1, · · · , φm) : X → R m . We study the set of ground states GS(α) of the potential α · Φ as a function of the direction vector α ∈ S m−1 . We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of Φ. In particular, for each α the set of rotation vectors of GS(α) forms a non-empty, compact and connected subset of a face Fα(Φ) of the rotation set associated with α. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in Fα(Φ). We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any m ∈ N examples with an exposed boundary point (i.e. Fα(Φ) being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face Fα(Φ) with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of GS(α) is a non-trivial line segment.

A shift map with a discontinuous entropy function

“…This follows from a more general analyticity result for so-called STP-maps (including SFT's, uniformly hyperbolic systems and expansive homeomorphisms with specification) and for Hölder continuous potentials, see [2,6,10]. We note that the reason for formulating our theorem for one-sided shift maps on a shift space with 3 symbols is for 1 We note that the theorems in [9,10,11,12] do not rely on the continuity of the localized entropy function. The only exception is Theorem A in [12] whose proof uses the continuity of H restricted to a line segment, i.e., m = 1.…”

“…One might suspect that the latter actually even guarantees the continuity of the localized entropy function for all dimensions m. Indeed, it was stated by Jenkinson [9, p. 3723] that the upper-semi continuity of the entropy map implies the continuity of the localized entropy. This claim was restated by Kucherenko and Wolf in [10,11,12]. 1 However, it turns out that the argument in [9] is incomplete.…”

On the computability of rotation sets and their entropies

“…Using techniques similar to those in the proofs for Example 2 in [41], one can show that Rot(Φ) = Conv{w(0), w(∞), w i (j) : j ≥ λ, i = 1, 2}.…”

“…Furthermore, by requiring additional properties on ℓ 1 , it can be arranged that w(∞) a smooth boundary point. We refer the reader to [41] for details. Next, we define approximations of the potential Φ.…”

“…Rotation sets play a role in several areas of ergodic theory and dynamical systems, and they have been studied recently by several authors, see, e.g., [6,5,22,26,25,33,38,40,42,64]. These studies include applications to higher-dimensional multifractal analysis, see, e.g., [1] and the references therein, ergodic optimization [22,34], and the study of ground states and zero-temperature measures [41].…”