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Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks

Let f be an orientation-preserving circle diffeomorphism with irrational “rotation number” of bounded type and finite number of break points, that is, the derivative f ′ has discontinuities of first kind at these points. Suppose f ′ satisfies a certain Zygmund condition which be dependent on para...

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Main Authors: Habibulla Akhadkulov,, Mohd. Salmi Md. Noorani,, Sokhobiddin Akhatkulov,
Format: Article
Language:English
Published: Penerbit Universiti Kebangsaan Malaysia 2014
Online Access:http://journalarticle.ukm.my/8610/1/jqma-10-2-paper8.pdf
http://journalarticle.ukm.my/8610/
http://www.ukm.my/jqma/jqma10_2a.html
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spelling my-ukm.journal.86102016-12-14T06:47:42Z http://journalarticle.ukm.my/8610/ Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks Habibulla Akhadkulov, Mohd. Salmi Md. Noorani, Sokhobiddin Akhatkulov, Let f be an orientation-preserving circle diffeomorphism with irrational “rotation number” of bounded type and finite number of break points, that is, the derivative f ′ has discontinuities of first kind at these points. Suppose f ′ satisfies a certain Zygmund condition which be dependent on parameter γ > 0 on each continuity intervals. We prove that the Rauzy-Veech renormalisations of f are approximated by Mobius transformations in C1 -norm if γ ∈(0,1] and in C2 -norm if γ ∈(1,∞) . In particular, we show that if f has zero mean nonlinearity, renormalisation of such maps approximated by piecewise affine interval exchange maps. Further, we consider two circle homeomorphisms with the same irrational “rotation number” of bounded type and finite number of break points. We prove that if they are not break equivalent then the conjugating map between these two maps is singular. Penerbit Universiti Kebangsaan Malaysia 2014-12 Article PeerReviewed application/pdf en http://journalarticle.ukm.my/8610/1/jqma-10-2-paper8.pdf Habibulla Akhadkulov, and Mohd. Salmi Md. Noorani, and Sokhobiddin Akhatkulov, (2014) Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks. Journal of Quality Measurement and Analysis, 10 (2). pp. 87-98. ISSN 1823-5670 http://www.ukm.my/jqma/jqma10_2a.html
institution Universiti Kebangsaan Malaysia
building Perpustakaan Tun Sri Lanang Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Kebangsaan Malaysia
content_source UKM Journal Article Repository
url_provider http://journalarticle.ukm.my/
language English
description Let f be an orientation-preserving circle diffeomorphism with irrational “rotation number” of bounded type and finite number of break points, that is, the derivative f ′ has discontinuities of first kind at these points. Suppose f ′ satisfies a certain Zygmund condition which be dependent on parameter γ > 0 on each continuity intervals. We prove that the Rauzy-Veech renormalisations of f are approximated by Mobius transformations in C1 -norm if γ ∈(0,1] and in C2 -norm if γ ∈(1,∞) . In particular, we show that if f has zero mean nonlinearity, renormalisation of such maps approximated by piecewise affine interval exchange maps. Further, we consider two circle homeomorphisms with the same irrational “rotation number” of bounded type and finite number of break points. We prove that if they are not break equivalent then the conjugating map between these two maps is singular.
format Article
author Habibulla Akhadkulov,
Mohd. Salmi Md. Noorani,
Sokhobiddin Akhatkulov,
spellingShingle Habibulla Akhadkulov,
Mohd. Salmi Md. Noorani,
Sokhobiddin Akhatkulov,
Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
author_facet Habibulla Akhadkulov,
Mohd. Salmi Md. Noorani,
Sokhobiddin Akhatkulov,
author_sort Habibulla Akhadkulov,
title Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
title_short Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
title_full Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
title_fullStr Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
title_full_unstemmed Notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
title_sort notes on conjugacies and renormalisations of circle diffeomorphisms with breaks
publisher Penerbit Universiti Kebangsaan Malaysia
publishDate 2014
url http://journalarticle.ukm.my/8610/1/jqma-10-2-paper8.pdf
http://journalarticle.ukm.my/8610/
http://www.ukm.my/jqma/jqma10_2a.html
_version_ 1643737528393531392
score 13.211869

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