On the measures of large entropy on a positive closed current

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