On the measures of large entropy on a positive closed current

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

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