Localized Pressure and Equilibrium States

Abstract

We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential ϕ is computed by considering only those (n, )-separated sets whose statistical sums with respect to an m-dimensional potential are "close" to a given value w ∈ R m . We then establish for several classes of systems and potentials ϕ and a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Hölder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, we construct an example with infinitely many ergodic localized equilibrium states. We also show that for systems with strong thermodynamic properties and w in the interior of the rotation set of there is at least one and at most finitely many localized equilibrium states.

The calculus of thermodynamical formalism

“…The second result states existence and uniqueness of equilibrium states under linear constraints; it is very close to Theorem B of [KW13], but even disregarding the difference in our hypotheses we obtain a more precise description of the equilibrium state: the parameter s of [KW13] is always equal to 1. In other words:…”

“…Theorem G notably shows that when T is the shift map and the test functions and the potential B all depend only on n coordinates, so does the potential of the constrained equilibrium state, which is thus a (n − 1)-Markovian measure (Remark 7.17, which also follows from the results of [KW13] but is not stated there).…”

“…Our goal here is to optimize the P B functionals (for example, the entropy h X ) on natural subsets of invariant measures, obtained by constraining the integrals of some functions. These questions have been considered by Jenkinson [Jen01] in the case of entropy and Kucherenko and Wolf [KW14, KW13], with somewhat different assumptions and methods. We believe that part of our claims are more explicit in some issues.…”

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

“…The number H(w) is also called localized entropy of w (see [19]). It follows from the upper-semi continuity of µ → h µ (f ) that the supremum in ( 4) is attained by at least one invariant measure and we call such a measure µ a localized measure of maximal entropy at w. Further, the map w → H(w) is continuous on Rot(Φ), see [18].…”

Nonlinear thermodynamical formalism

“…As pointed out by one of the referees, nonlinear energies have been considered in the distinct but related "multifractal analysis" (see Remark 3.5, and [Cli14] and the references therein for background). Theorem C is also related to constrained equilibrium measures as considered in [KW15] and [GKLMF18], see Remark 3.18.…”