Dynamics of covering maps of the annulus I: semiconjugacies

TL;DR

This paper explores when covering maps of the annulus can be semiconjugate to circle maps, analyzing the implications for classification and identifying cases where such semiconjugacies do not exist, thus raising new questions.

Contribution

It investigates the conditions under which annulus covering maps are semiconjugate to circle maps and discusses the implications for their classification.

Findings

Some annulus covering maps are not semiconjugate to circle maps.

Semiconjugacy influences the classification of annulus maps.

Open questions arise from examples lacking semiconjugacy.

Abstract

It is often the case that a covering map of the open annulus is semiconjugate to a map of the circle of the same degree. We investigate this possibility and its consequences on the dynamics. In particular, we address the problem of the classification up to conjugacy. However, there are examples which are not semiconjugate to a map of the circle, and this opens new questions.