On the dimension of invariant measures of endomorphisms of CPᵏ

الأصل · EN

Let f be an endomorphism of CPᵏ 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 H μ≥ d λ₁ + d λ₂, 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 CPᵏ. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in CPᵏ and a normalization theorem for the ν-generic inverse branches of fⁿ.