Strangely dispersed minimal sets in the quasiperiodically forced Arnold map

TL;DR

This paper investigates quasiperiodically forced circle endomorphisms, revealing the existence of uncountably many complex minimal sets and generalizing key properties of circle endomorphisms to the forced case, with implications for the Arnold map.

Contribution

It introduces the concept of 'strangely dispersed' minimal sets in quasiperiodically forced systems and extends classical results to this more complex setting.

Findings

Existence of uncountably many minimal sets with complex structure

All rotation numbers in the rotation interval are realized on minimal sets

Positive topological entropy when the rotation interval has non-empty interior

Abstract

We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as `strangely dispersed'. Along the way, we generalise some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realised on minimal sets, and if the rotation interval has non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example.