MAXIMIZING MEASURES ON METRIZABLE NON-COMPACT SPACES

Abstract

AbstractWe prove the existence and uniqueness of maximizing measures for various classes of continuous integrands on metrizable (non-compact) spaces and close subsets of Borel probability measures. We apply these results to various dynamical contexts, especially to hyperbolic mappings of the form $f_\lambda(z)=\lambda\mathrm{e}^z$, $\lambda\ne0$, and associated canonical maps $F_\lambda$ of an infinite cylinder. It is then shown that, for all hyperbolic maps $F_\lambda$, all dynamically maximizing measures have compact supports and, for all $0^+$-potentials $\phi$, the set of (weak) limit points of equilibrium states of potentials $t\phi$, $t\nearrow+\infty$, is non-empty and consists of dynamically maximizing measures.