On the dimension of invariant measures of endomorphisms of $${mathbb{C}mathbb{P}^k}$$

Let f be an endomorphism of mathbbCmathbbP^k{mathbb{C}mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{rm dim}_mathcal{H}mu geq {{rm log} d over lambda_1} + {{rm 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 ν-generic inverse branches of f ⁿ in mathbbCmathbbP^k{mathbb{C}mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in mathbbCmathbbP^k{mathbb{C}mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .