Lyapunov minimizing measures for expanding maps of the circle

Abstract

We consider the set of maps f\in\mathcal{F}_{\alpha+} = \cup_{\beta>\alpha} \mathcal{C}^{1+\beta} of the circle which are covering maps of degree D, expanding, \min_{x\in S^1}f'(x) >1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f-invariant probability measure \mu minimizing \int \ln f'\,d\mu over all f-invariant probability measures. We show there exists a set \mathcal{G}_+\subset\mathcal{F}_{\alpha+}, open and dense in the \mathcal{C}^{1+\alpha}-topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the \mathcal{C}^{1+\alpha}-topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.

We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous quation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8].

We will also present some results on the set of f-invariant measures \mu maximizing \int A\,d\mu for a fixed \mathcal{C}^1-expanding map f and a general potential A, not necessarily equal to -\ln f'.