Derivatives of rotation number of one parameter families of circle diffeomorphisms

TL;DR

This paper investigates the behavior of the rotation number function for a family of circle diffeomorphisms, establishing a lower bound on its derivative when the rotation number is irrational.

Contribution

It provides a new lower bound on the derivative of the rotation number function for irrational values, enhancing understanding of its regularity properties.

Findings

The derivative of the rotation number function is at least 1 for irrational rotation numbers.

The paper establishes a limsup inequality related to the derivatives of the rotation number.

It advances the theoretical understanding of the smoothness and variation of rotation numbers in circle dynamics.

Abstract

We consider the rotation number $\rho(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $\rho(t)$ is irrational $$\limsup_{t'\to t}(\rho(t')-\rho(t))/(t'-t)\geq 1.$$