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

Abstract

Let f be an endomorphism of \({\mathbb{C}\mathbb{P}^k}\) and ν be an f-invariant measure with positive Lyapunov exponents (λ1, . . . , λk). 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 \({{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}\), 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 fn in \({\mathbb{C}\mathbb{P}^k}\) . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \({\mathbb{C}\mathbb{P}^k}\) and a normalization theorem for the ν-generic inverse branches of fn.