Dimensional preimage entropies

TL;DR

This paper introduces a generalized form of topological entropy for complex manifolds, measuring the dynamics of meromorphic maps on local analytic sets of various dimensions, and relates it to Lyapunov exponents.

Contribution

It defines a new entropy-like quantity for meromorphic maps on complex manifolds and establishes inequalities linking it to Lyapunov exponents.

Findings

Introduces $h_{(m,l)}^{top}(f)$ as a generalized entropy measure.

Establishes inequalities between this entropy and Lyapunov exponents.

Extends classical entropy concepts to local analytic sets in complex dynamics.

Abstract

Let $X$ be a compact complex manifold of dimension $k$ and $f:X \longrightarrow X$ be a dominating meromorphic map. We generalize the notion of topological entropy, by defining a quantity $h_{(m,l)}^{top}(f)$ which measures the action of $f$ on local analytic sets $W$ of dimension $l$ with $W \subset f^{-n}(\Delta)$ where $\Delta$ is a local analytic set of dimension $m$. We give then inequalities between $h_{(m,l)}^{top}(f)$ and Lyapounov exponents of suitable invariant measures.