On the measures of large entropy on a positive closed current

Abstract

Let $f:X\to X$ be a dominating meromorphic map of a compact K\"ahler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $\nu$ of large entropy supported by $\mathrm{supp}(S)$. Defining dimensions for $\nu$ and $S$, we give inequalities \`a la Ma\~n\'e involving the Lyapunov exponents of $\nu$ and those dimensions. We give dynamical applications of those inequalities.