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We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential $varphi$ is computed by considering only those $(n,epsilon)$-separated sets whose statistical sums with respect to an $m$-dimensional potential $Phi$ are close to a given value $win bR^m$. We then establish for several classes of systems and potentials $varphi$ and $Phi$ 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 Holder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, ergodic localized equilibrium states for Holder continuous potentials are in general not unique.
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Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.
Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphsim $f$. We define the u-pressure $P^u(f, varphi)$ of $f$ at a continuous function $varphi$ via the dynamics of $f$ on local unstable leaves. A variational principle for unstable pressure $P^u(f, varphi)$, which states that $P^u(f, varphi)$ is the supremum of the sum of the unstable entropy and the integral of $varphi$ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fr{e}chet differentiability and their relations to u-equilibrium states, are also considered.
Pfister and Sullivan proved that if a topological dynamical system $(X,T)$ satisfies almost product property and uniform separation property, then for each nonempty compact %convex subset $K$ of invariant measures, the entropy of saturated set $G_{K}$ satisfies begin{equation}label{Bowens topological entropy} h_{top}^{B}(T,G_{K})=inf{h(T,mu):muin K}, end{equation} where $h_{top}^{B}(T,G_{K})$ is Bowens topological entropy of $T$ on $G_{K}$, and $h(T,mu)$ is the Kolmogorov-Sinai entropy of $mu$. In this paper, we investigate topological complexity of $G_{K}$ by replacing Bowens topological entropy with upper capacity entropy and packing entropy and obtain the following formulas: begin{equation*} h_{top}^{UC}(T,G_{K})=h_{top}(T,X) mathrm{and} h_{top}^{P}(T,G_{K})=sup{h(T,mu):muin K}, end{equation*} where $h_{top}^{UC}(T,G_{K})$ is the upper capacity entropy of $T$ on $G_{K}$ and $h_{top}^{P}(T,G_{K})$ is the packing entropy of $T$ on $G_{K}.$ In the proof of these two formulas, uniform separation property is unnecessary.
In this paper we introduce the definition of topological $r$-pressure of free semigroup actions on compact metric space and provide some properties of it. Through skew-product transformation into a medium, we can obtain the following two main results. 1. We extend the result that the topological pressure is the limit of topological $r$-pressure incite{C} to free semigroup actions ($rto 0$). 2. Let $f_i,$ $i=0, 1, cdots, m-1$, be homeomorphisms on a compact metric space. For any continuous function, we verify that the topological pressure of $f_0, cdots, f_{m-1}$ equals the topological pressure of $f_0^{-1}, cdots, f_{m-1}^{-1}.$
We explore an approach to the conjecture of Katok on intermediate entropies that based on uniqueness of equilibrium states, provided the entropy function is upper semi-continuous. As an application, we prove Katoks conjecture for Ma~ne diffeomorphisms.
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