Geometry and entropy of generalized rotation sets
Time and place
1 PM on Thursday, November 8th, 2012; NAC 6/1113
Tamara Kucherenko (CCNY)
Abstract
For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and $\R(\Phi)$ is the set of all $\mu$-integrals of $\Phi$ and $\mu$ runs over all $f$-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of $\bR^m$. We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set $K$ in $\bR^m$ a potential $\Phi=\Phi(K)$ with $\R(\Phi)=K$. Next, we study the relation between $\R(\Phi)$ and the set of all statistical limits $\R_{Pt}(\Phi)$. We show that in general these sets differ but also provide criteria that guarantee $\R(\Phi)= \R_{Pt}(\Phi)$. Finally, we study the entropy function $w\mapsto H(w), w\in \R(\Phi)$. and establish a variational principle for the entropy function.
