Attracting Currents and Equilibrium Measures for Quasi-attractors of $\mathbb P^k$

TL;DR

This paper constructs attracting currents and equilibrium measures for quasi-attractors of holomorphic endomorphisms on complex projective space, providing a new ergodic approach and insights into their entropy and bifurcations.

Contribution

It introduces a systematic method to associate equilibrium measures to quasi-attractors, enabling a deeper ergodic and bifurcation analysis of these complex dynamical objects.

Findings

Existence of at most countably many quasi-attractors with entropy multiples of log d

Construction of finite sets of attracting currents for each quasi-attractor

Potential development of a bifurcation theory for attracting sets

Abstract

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of degree $d.$ For each quasi-attractor of $f$ we construct a finite set of currents with attractive behaviors. To every such an attracting current is associated an equilibrium measure which allows for a systematic ergodic theoretical approach in the study of quasi-attractors of $\mathbb P^k.$ As a consequence, we deduce that there exist at most countably many quasi-attractors, each one with topological entropy equal to a multiple of $\log d.$ We also show that the study of these analytic objects can initiate a bifurcation theory for attracting sets.