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$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

Bertuel Tangue Ndawa, Carlos Ogouyandjou

Abstract

We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological conjugacy between two maps of my class is a bi-Lipschitz homeomorphism, then it is a $C^1$ diffeomorphism; that is, the bi-Lipschitz homeomorphism class and $C^1$ diffeomorphism class of a map in our class are equivalent.

$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

Abstract

We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological conjugacy between two maps of my class is a bi-Lipschitz homeomorphism, then it is a diffeomorphism; that is, the bi-Lipschitz homeomorphism class and diffeomorphism class of a map in our class are equivalent.

Paper Structure

This paper contains 9 sections, 6 theorems, 34 equations.

Table of Contents

  1. Abstract
  2. Introduction
  3. The class of functions:
  4. Notations.
  5. Common notations.
  6. Parameters frequently used.
  7. Main result
  8. Renormalization
  9. $C^1$ diffeomorphism class

Key Result

Proposition 1

Let $({{\ell_1} },{{\ell_2} })\in(1,2)^2$. Then, there exists $\lambda_u>1,\;\lambda_s\in(0,1)\; E^u,\;E^s$, $E^+,$$w_{fix}\in{\mathbb R}^4,$ such that the following holds. Given $f\in \widetilde{\mathcal{W}}^X_{[1]}$ with critical exponents $({{\ell_1} },{{\ell_2} })$, there exists $c_u(f), c'_u(f and where, $O(e,c_u(f),\lambda_u,n)$ is the vector whose components are with, And, we have also

Theorems & Definitions (10)

  • Remark 1
  • Proposition 1: Proposition 12, NTBr
  • Lemma 1: Lemma 20, NTBr
  • Proposition 2
  • proof
  • Lemma 2: Lemma 26, NTBr
  • Lemma 3
  • proof
  • Proposition 3
  • proof