# Rigidity of critical circle mappings I

Edson de Faria; Welington de Melo

Journal of the European Mathematical Society (1999)

- Volume: 001, Issue: 4, page 339-392
- ISSN: 1435-9855

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topde Faria, Edson, and de Melo, Welington. "Rigidity of critical circle mappings I." Journal of the European Mathematical Society 001.4 (1999): 339-392. <http://eudml.org/doc/277440>.

@article{deFaria1999,

abstract = {We prove that two $C^3$ critical circle maps with the same rotation number in a special set $\mathbb \{A\}$ are $C^\{1+\alpha \}$ conjugate for some $\alpha >0$ provided their successive renormalizations
converge together at an exponential rate in the $C^0$ sense. The set $\mathbb \{A\}$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of
$C^\infty $ critical circle maps with the same rotation number that are not $C^\{1+\beta \}$ conjugate for any $\beta >0$. The class of rotation numbers for which such examples exist contains Diophantine numbers.},

author = {de Faria, Edson, de Melo, Welington},

journal = {Journal of the European Mathematical Society},

keywords = {critical circle map; rotation number; Diophantine numbers; circle automorphisms; renormalization; rotation numbers; smooth conjugacy},

language = {eng},

number = {4},

pages = {339-392},

publisher = {European Mathematical Society Publishing House},

title = {Rigidity of critical circle mappings I},

url = {http://eudml.org/doc/277440},

volume = {001},

year = {1999},

}

TY - JOUR

AU - de Faria, Edson

AU - de Melo, Welington

TI - Rigidity of critical circle mappings I

JO - Journal of the European Mathematical Society

PY - 1999

PB - European Mathematical Society Publishing House

VL - 001

IS - 4

SP - 339

EP - 392

AB - We prove that two $C^3$ critical circle maps with the same rotation number in a special set $\mathbb {A}$ are $C^{1+\alpha }$ conjugate for some $\alpha >0$ provided their successive renormalizations
converge together at an exponential rate in the $C^0$ sense. The set $\mathbb {A}$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of
$C^\infty $ critical circle maps with the same rotation number that are not $C^{1+\beta }$ conjugate for any $\beta >0$. The class of rotation numbers for which such examples exist contains Diophantine numbers.

LA - eng

KW - critical circle map; rotation number; Diophantine numbers; circle automorphisms; renormalization; rotation numbers; smooth conjugacy

UR - http://eudml.org/doc/277440

ER -

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