  Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $\nu$ be an $f$-invariant
measure with positive Lyapunov exponents $(\lambda_1,\...,\lambda_k)$. We prove
a lower bound for the pointwise dimension of $\nu$ in terms of the degree of
$f$, the exponents of $\nu$ and the entropy of $\nu$. In particular our result
can be applied for the maximal entropy measure $\mu$. When $k=2$, it implies
that the Hausdorff dimension of $\mu$ is estimated by $\dim_{\cal H} \mu \geq
{\log d \over \lambda_1} + {\log d \over \lambda_2}$, which is half of the
conjectured formula. Our method for proving these results consists in studying
the distribution of the $\nu$-generic inverse branches of $f^n$ in
$\mathbb{CP}^k$. Our tools are a volume growth estimate for the bounded
holomorphic polydiscs in $\mathbb{CP}^k$ and a normalization theorem for the
$\nu$-generic inverse branches of $f^n$.
