Friday, March 4, 2016

Equilibrium States for Hyperbolic Potentials

Vanessa Ramos
(CMUP and Universidade Federal do Maranhão)

Abstract: Let $f:M\rightarrow M$ be a continuous map defined on a compact metric space $M$ and let $\phi:M\rightarrow\mathbb{R}$ be a real continuous function. In this classical setting, we say that $\mu_{\phi}$ is an equilibrium state associated to $(f,\phi)$, if $\mu_{\phi}$ is an $f$-invariant probability measure characterized by the following variational principle: $$ P_{f}(\phi)=h_{\mu_{\phi}}(f)+\displaystyle\int{\!\phi} \,d\mu_{\phi}=\sup_{\mu\in\mathcal{M}_{f}(M)}\left\{h_{\mu}(f)+\int{\!\phi}\, d\mu\right\}$$ where $P_{f}(\phi)$ denotes the topological pressure, $h_{\mu}(f)$ is the metric entropy and the supremum is taken over all $f$-invariants probabilities measures. Existence is a relatively soft property that can often be established via compactness arguments. Uniqueness is usually more subtle and requires a better understanding of the dynamics. In this talk, we will show uniqueness of equilibrium states associated to local diffeomorphisms $f:M\rightarrow M$ and hyperbolic Holder continuous functions $\phi:M\rightarrow\mathbb{R}$.

Friday, March 4, 2016
Time: 14h30
Room: C6.2.33, Faculdade de Ciências, UL