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Register a new userPressure and Equilibrium States in Ergodic Theory
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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.
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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.
We show that every totally ergodic generalised matrix equilibrium state is psi-mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.
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.
We survey the impact of the Poincare recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory.
This survey is an update of the 2008 version, with recent developments and new references.
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