Limit Drift for Complex Feigenbaum Mappings

TL;DR

This paper investigates the complex dynamics of Feigenbaum polynomial towers, focusing on measure properties, invariant measures, and drift convergence as the critical point order increases, revealing new insights into their limiting behavior.

Contribution

It establishes the existence and properties of invariant measures for Feigenbaum towers and demonstrates the convergence of drifts to a finite limit as the critical point order tends to infinity.

Findings

Invariant measures exist and are absolutely continuous for tower dynamics.

Drifts converge to a finite limit expressed via the limiting tower.

Julia set measure positivity relates to drift sign and tower dynamics.

Abstract

We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order \ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding \ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when \ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point