Topology of the regular part for infinitely renormalizable quadratic polynomials

TL;DR

This paper investigates the topological structure of the inverse limits of infinitely renormalizable quadratic polynomials, demonstrating rigidity under certain bounds and advancing understanding of their natural extensions.

Contribution

It provides a detailed topological description of the inverse limit space for these polynomials and establishes rigidity results under a-priori bounds.

Findings

Topology of inverse limits is rigid modulo combinatorics for bounded renormalizations

Describes natural extensions of quadratic polynomials in the renormalization context

Advances understanding of the structure of infinitely renormalizable quadratic polynomials

Abstract

In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics.