On the abundance of $k$-fold semi-monotone minimal sets in bimodal circle maps

TL;DR

This paper investigates the structure and abundance of semi-monotone minimal sets in bimodal circle maps, revealing a rich geometric and symbolic structure related to rotation numbers and invariant measures.

Contribution

It extends the understanding of invariant sets in bimodal circle maps by characterizing their abundance and structure for both irrational and rational rotation numbers.

Findings

Existence of a (k-1)-dimensional ball of invariant measures for irrational rotation numbers.

Complete description of periodic orbit analogs for rational rotation numbers.

Generalization of Hedlund and Morse construction for symbolic analysis.

Abstract

Inspired by a twist maps theorem of Mather we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map $g$ to the $k$-fold cover. For each irrational in the interior of the rotation set the collection of the $k$-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball in the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool is a generalization of a construction of Hedlund and Morse which generates the symbolic analogs of these $k$-fold well-ordered invariant sets.