Cyclicity in families of circle maps

TL;DR

This paper investigates the number of periodic trajectories in families of circle maps with small perturbations, focusing on how typical functions influence cyclicity.

Contribution

It provides new insights into the cyclicity of circle maps under small perturbations for typical functions.

Findings

Quantifies the maximum number of periodic trajectories for small parameter values.

Shows how the structure of the function $f$ affects cyclicity.

Establishes conditions for typical functions to have certain cyclicity properties.

Abstract

In this paper we will study families of circle maps of the form $x\mapsto x+2\pi r + a f(x) \pmod{2\pi}$ and investigate how many periodic trajectories maps from this family can have for a "typical" function $f$ provided the parameter $a$ is small.