Renormalizations of circle hoemomorphisms with a single break point

Abdumajid Begmatov; Akhtam Dzhalilov; Dieter Mayer

doi:10.3934/dcds.2014.34.4487

Article Contents

2014, Volume 34, Issue 11: 4487-4513. Doi: 10.3934/dcds.2014.34.4487

This issue Previous Article Topological and ergodic properties of symmetric sub-shifts Next Article Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation

Renormalizations of circle hoemomorphisms with a single break point

1.
Faculty of Mathematics and Mechanics. Samarkand State University, Boulevard Street 15, 140104 Samarkand
2.
Turin Politechnic University in Tashkent, Kichik halqa yuli 17, 100095 Tashkent
3.
Institut für Theoretische Physik, TU Clausthal, Leibnizstrasse 10, D-38678 Clausthal-Zellerfeld

Received: October 31, 2013

Revised: February 28, 2014

Published: October 2014

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Abstract

Let $f$ be an orientation preserving circle homeomorphism with a single break point $x_b,$ i.e. with a jump in the first derivative $f'$ at the point $x_b,$ and with irrational rotation number $\rho=\rho_{f}.$ Suppose that $f$ satisfies the Katznelson and Ornstein smoothness conditions, i.e. $f'$ is absolutely continuous on $[x_b,x_b+1]$ and $f''(x)\in \mathbb{L}^{p}([0,1), d\ell)$ for some $p>1$, where $\ell$ is Lebesque measure. We prove, that the renormalizations of $f$ are approximated by linear-fractional functions in $\mathbb{C}^{1+L^{1}}$, that means, $f$ is approximated in $C^{1}-$ norm and $f''$ is appoximated in $L^{1}-$ norm. Also it is shown, that renormalizations of circle diffeomorphisms with irrational rotation number satisfying the Katznelson and Ornstein smoothness conditions are close to linear functions in $\mathbb{C}^{1+L^{1}}$- norm.

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Mathematics Subject Classification: Primary: 37E10, 37E20; Secondary: 37C15, 37C40.

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References

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