Rigidity of critical circle mappings I

de 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 -