Entropy of meromorphic maps acting on analytic sets

Let $f : Xto X$ be a dominating meromorphic map on a compact Kahler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0leq l leq k$. For an ergodic probability measure $ u$, we extend similarly the notion of measure-theoretic entropy $h_{ u}^l(f)$. Under mild hypothesis, we compute $h^l_{mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $mathbb{P}^2$ of degree $d$, we show that $h^1_{mathrm{top}}(f)= log d$ for a large class of maps but we give examples where $h^1_{mathrm{top}}(f) eq log d$.