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

TL;DR

This paper establishes a lower bound for the pointwise dimension of invariant measures of endomorphisms of complex projective space, linking it to degree, entropy, and Lyapunov exponents, with implications for Hausdorff dimension estimates.

Contribution

It introduces a novel method analyzing inverse branches to derive dimension bounds for invariant measures on CP^k, advancing understanding of their geometric properties.

Findings

Lower bound for pointwise dimension in terms of degree, entropy, and Lyapunov exponents.

For k=2, Hausdorff dimension estimate matches a specific formula involving log d and Lyapunov exponents.

Method employs volume growth estimates and normalization of inverse branches.

Abstract

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 $f^n$.