Mapping schemes realizable by obstructed topological polynomials

TL;DR

This paper explores the realizability of obstructed topological polynomials, demonstrating that certain non-hyperbolic post-critical dynamics cannot be realized by complex polynomials, thus extending understanding of polynomial obstructions.

Contribution

It provides a partial converse to the Berstein-Levy Theorem by showing the existence of obstructed topological polynomials for specific non-hyperbolic dynamics.

Findings

Existence of obstructed topological polynomials for certain dynamics

Use of self-similar groups to identify obstructions

Construction of numerous examples of obstructed polynomials

Abstract

In 1985, Levy used a theorem of Berstein to prove that all hyperbolic topological polynomials are equivalent to complex polynomials. We prove a partial converse to the Berstein-Levy Theorem: given post-critical dynamics that are in a sense strongly non-hyperbolic, we prove the existence of topological polynomials which are not equivalent to any complex polynomial that realize these post-critical dynamics. This proof employs the theory of self-similar groups to demonstrate that a topological polynomial admits an obstruction and produces a wealth of examples of obstructed topological polynomials.