no. 3
Renormalization for piecewise smooth homeomorphisms on the circle
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 441-462.

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy–Veech renormalizations of generalized interval exchange maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converge to the set of piecewise affine interval exchange maps.

DOI : 10.1016/j.anihpc.2012.09.004
Classification : 37E10, 37E05, 37E20, 37C05, 37B10
Mots clés : Renormalization, Interval exchange transformations, Rauzy–Veech induction, Universality, Homeomorphism on the circle, Convergence 
@article{AIHPC_2013__30_3_441_0,
     author = {Cunha, Kleyber and Smania, Daniel},
     title = {Renormalization for piecewise smooth homeomorphisms on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {441--462},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.09.004},
     zbl = {1338.37052},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.09.004/}
}
TY  - JOUR
AU  - Cunha, Kleyber
AU  - Smania, Daniel
TI  - Renormalization for piecewise smooth homeomorphisms on the circle
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 441
EP  - 462
VL  - 30
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.09.004/
DO  - 10.1016/j.anihpc.2012.09.004
LA  - en
ID  - AIHPC_2013__30_3_441_0
ER  - 
%0 Journal Article
%A Cunha, Kleyber
%A Smania, Daniel
%T Renormalization for piecewise smooth homeomorphisms on the circle
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 441-462
%V 30
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.09.004/
%R 10.1016/j.anihpc.2012.09.004
%G en
%F AIHPC_2013__30_3_441_0
Cunha, Kleyber; Smania, Daniel. Renormalization for piecewise smooth homeomorphisms on the circle. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 441-462. doi : 10.1016/j.anihpc.2012.09.004. http://www.numdam.org/articles/10.1016/j.anihpc.2012.09.004/

[1] K. Cunha, Transformações de intercâmbio de intervalos generalizadas de genus 1, Ph.D. Thesis, ICMC-USP, Brazil, 2010.

[2] K. Cunha, D. Smania, Rigidity for piecewise smooth homeomorphisms on the circle, preprint, 2012, http://lanl.arxiv.org/abs/1201.1401.

[3] Welington De Melo, Sebastian Van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. (3) vol. 25, Springer-Verlag, Berlin (1993) | Zbl

[4] M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5-233 | EuDML | Numdam | Zbl

[5] M. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25-31 | EuDML | Zbl

[6] Y. Katznelson, D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 no. 4 (1989), 643-680 | Zbl

[7] K.M. Khanin, Ya.G. Sinaĭ, A new proof of M. Hermanʼs theorem, Comm. Math. Phys. 112 no. 1 (1987), 89-101 | Zbl

[8] K. Khanin, A. Teplinsky, Hermanʼs theory revisited, Invent. Math. 178 no. 2 (2009), 333-344 | Zbl

[9] K.M. Khanin, E.B. Vul, Circle homeomorphisms with weak discontinuities, Dynamical Systems and Statistical Mechanics, Moscow, 1991, Adv. Soviet Math. vol. 3, Amer. Math. Soc., Providence, RI (1991), 57-98 | Zbl

[10] M. Martens, The periodic points of renormalization, Ann. of Math. (2) 147 no. 3 (1998), 543-584 | Zbl

[11] A. Nogueira, D. Rudolph, Topological weak-mixing of interval exchange maps, Ergodic Theory Dynam. Systems 17 no. 5 (1997), 1183-1209 | Zbl

[12] G. Rauzy, Échanges dʼintervalles et transformations induites, Acta Arith. 34 no. 4 (1979), 315-328 | EuDML | Zbl

[13] Ya.G. Sinaĭ, Topics in Ergodic Theory, Princeton Math. Ser. vol. 44, Princeton University Press, Princeton, NJ (1994) | Zbl

[14] Ya.G. Sinaĭ, K.M. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Uspekhi Mat. Nauk 44 no. 1(265) (1989), 57-82 | Zbl

[15] W.A. Veech, Interval exchange transformations, J. Anal. Math. 33 (1978), 222-272 | Zbl

[16] W.A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 no. 1 (1982), 201-242 | Zbl

[17] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4) 17 no. 3 (1984), 333-359 | EuDML | Numdam | Zbl

Cité par Sources :