  Let $f : X\to X$ be a dominating meromorphic map on a compact K\"ahler
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 $0\leq l \leq k$. For an ergodic probability measure $\nu$, we extend
similarly the notion of measure-theoretic entropy $h_{\nu}^l(f)$.
  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 $h^1_{\mathrm{top}}(f)\neq
\log d$.
