at $0$.
address: 'Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France.'
author:
- Kuntal Banerjee
title: 'Boundaries of the Arnol’d tongues and the standard family'
---

Mathematics Subject Classes : 37E10; 26A18; 30D05

Introduction
============

In this article we study certain structures in the parameter space of families of circle homeomorphisms. Poincaré was the first person to study the dynamics of circle homeomorphisms while he was looking at the solutions of differential equations on torus in his 1885 mémoire [@poincare]. Formally, if we start with an increasing homeomorphism $F:\R \to \R$ of the real line such that $F(x+1)=F(x)+1$ for any $x\in \R$, then $F$ will induce an orientation preserving circle homeomorphism $f: \T \to \T$ given by $[x] \mapsto [F(x)]$. Note that we are denoting the circle as the additive group $\T=\R/\Z$. A natural question is how much every point on the real line is translated on average under $F$ or how much every point on the circle is rotated on average under the action of $f$. If we want to look at the average displacement after the $n$-th iterate of $F$ at point $x\in \R$, then we should look at 