Generic families of circle diffeomorphisms have many coexisting periodic orbits

TL;DR

This paper demonstrates that generic families of circle diffeomorphisms exhibit dense parameter values with many coexisting periodic orbits near those with irrational rotation numbers, indicating a lack of weak structural stability.

Contribution

It establishes the prevalence of parameter values with arbitrarily many periodic orbits near irrational rotation numbers in generic circle diffeomorphism families.

Findings

Irrational rotation numbers are approximated by parameters with large finite periodic orbits.

Families with irrational rotation numbers are not weakly structurally stable.

A residual set of families yields a continuum of weak equivalence classes.

Abstract

We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which the diffeomorphisms have arbitrarily large finite numbers of periodic orbits. This phenomenon implies that families where irrational rotation numbers appear are not weakly structurally stable. Moreover, we prove that any locally residual set of one-parameter families with nonconstant rotation number yields a continuum of weak equivalence classes of families.