Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

TL;DR

This paper studies the dynamics of certain meromorphic maps on compact Kähler surfaces with small topological degree, focusing on translating cohomological data into invariant currents to understand their entropy and geometric properties.

Contribution

It develops a method to convert cohomological information into invariant currents, advancing the understanding of dynamics for meromorphic maps with small topological degree.

Findings

Established a link between cohomology action and invariant currents.

Provided examples of maps on irrational surfaces.

Identified properties of maps with vanishing self-intersection classes.

Abstract

We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection.