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Real bounds and Lyapunov exponents
1. | Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP, Brazil |
2. | Instituto de Matemática e Estatstica, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140, Niterói, Rio de Janeiro, Brazil |
References:
[1] |
A. Avila, On rigidity of critical circle maps, Bull. Braz. Math. Soc., 44 (2013), 611-619.
doi: 10.1007/s00574-013-0027-5. |
[2] |
A. Avila, X. Buff and A. Chéritat, Siegel disks with smooth boundaries, Acta Math., 193 (2004), 1-30.
doi: 10.1007/BF02392549. |
[3] |
H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.
doi: 10.1007/s00222-007-0108-4. |
[4] |
T. Clark and S. van Strien, Quasisymmetric rigidity in one-dimensional dynamics, manuscript, 2014. |
[5] |
M. Cortez and J. Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps, Israel. J. Math., 176 (2010), 157-193.
doi: 10.1007/s11856-010-0024-y. |
[6] |
M. Cortez and J. Rivera-Letelier, Choquet simplices as spaces of invariant probability measures on post-critical sets, Ann. Henri Poincaré, 27 (2010), 95-115.
doi: 10.1016/j.anihpc.2009.07.008. |
[7] |
A. Douady, Disques de Siegel et aneaux de Herman, Sém. Bourbaki 1986/87, Astérisque, 1986/87 (1987), 151-172. |
[8] |
G. Estevez and E. de Faria, Real bounds and quasisymmetric rigidity for multicritical circle maps, in preparation. |
[9] |
E. de Faria, Proof of Universality for Critical Circle Mappings, Ph.D. Thesis, CUNY, 1992. |
[10] |
E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergod. Th. & Dynam. Sys., 19 (1999), 995-1035.
doi: 10.1017/S0143385799133959. |
[11] |
E. de Faria and W. de Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc., 1 (1999), 339-392.
doi: 10.1007/s100970050011. |
[12] |
E. de Faria and W. de Melo, Rigidity of critical circle mappings II, J. Amer. Math. Soc., 13 (2000), 343-370.
doi: 10.1090/S0894-0347-99-00324-0. |
[13] |
E. de Faria, W. de Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math., 164 (2006), 731-824.
doi: 10.4007/annals.2006.164.731. |
[14] |
P. Guarino, Rigidity Conjecture for $C^3$ Critical Circle Maps, Ph.D. Thesis, IMPA, 2012, available at www.preprint.impa.br. |
[15] |
P. Guarino, M. Martens and W. de Melo, Rigidity of smooth critical circle maps: unbounded combinatorics, in preparation. |
[16] |
P. Guarino and W. de Melo, Rigidity of smooth critical circle maps, available at arXiv:1303.3470. |
[17] |
J. Guckenheimer, Sensitive Dependence to Initial Conditions for One Dimensional Maps, Commun. Math. Phys., 70 (1979), 133-160.
doi: 10.1007/BF01982351. |
[18] |
G. R. Hall, A $C^\infty$ Denjoy counterexample, Ergod. Th. & Dynam. Sys., 1 (1981), 261-272. |
[19] |
M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. |
[20] |
M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations, manuscript, 1988. |
[21] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[22] |
G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergod. Th. & Dynam. Sys., 10 (1990), 717-744.
doi: 10.1017/S0143385700005861. |
[23] |
K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities, Invent. Math., 169 (2007), 193-218.
doi: 10.1007/s00222-007-0047-0. |
[24] |
A. Ya. Khinchin, Continued Fractions, (reprint of the 1964 translation), Dover Publications, Inc., 1997. |
[25] |
D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps, Mosc. Math. J., 6 (2006), 317-351. |
[26] |
S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-1-4612-4220-8. |
[27] |
S. Li and W. Shen, Hausdorff dimension of Cantor attractors in one-dimensional dynamics, Invent. Math., 171 (2008), 345-387.
doi: 10.1007/s00222-007-0083-9. |
[28] |
W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-3-642-78043-1. |
[29] |
T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680.
doi: 10.1007/s002220050236. |
[30] |
C. L. Petersen, The Herman-Świątek theorems with applications, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 274 (2000), 211-225. |
[31] |
F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.
doi: 10.1090/S0002-9939-1993-1186141-9. |
[32] |
F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63.
doi: 10.1007/s00222-002-0243-x. |
[33] |
F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Scient. Éc. Norm. Sup., 40 (2007), 135-178.
doi: 10.1016/j.ansens.2006.11.002. |
[34] |
F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010.
doi: 10.1017/CBO9781139193184. |
[35] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps, available at arXiv:1204.3071. |
[36] |
D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), 417-466, Amer. Math. Soc., Providence, RI, 1992. |
[37] |
G. Świątek, Rational rotation numbers for maps of the circle, Commun. Math. Phys., 119 (1988), 109-128.
doi: 10.1007/BF01218263. |
[38] |
M. Yampolsky, Complex bounds for renormalization of critical circle maps, Ergod. Th. & Dynam. Sys., 19 (1999), 227-257.
doi: 10.1017/S0143385799120947. |
[39] |
M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Commun. Math. Phys., 218 (2001), 537-568.
doi: 10.1007/PL00005561. |
[40] |
M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. IHES, 96 (2002), 1-41.
doi: 10.1007/s10240-003-0007-1. |
[41] |
M. Yampolsky, Renormalization horseshoe for critical circle maps, Commun. Math. Phys., 240 (2003), 75-96.
doi: 10.1007/s00220-003-0891-8. |
[42] |
J.-C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C.R. Acad. Sc. Paris, 298 (1984), 141-144. |
[43] |
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks, Invent. Math., 185 (2011), 421-466.
doi: 10.1007/s00222-011-0312-0. |
show all references
References:
[1] |
A. Avila, On rigidity of critical circle maps, Bull. Braz. Math. Soc., 44 (2013), 611-619.
doi: 10.1007/s00574-013-0027-5. |
[2] |
A. Avila, X. Buff and A. Chéritat, Siegel disks with smooth boundaries, Acta Math., 193 (2004), 1-30.
doi: 10.1007/BF02392549. |
[3] |
H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.
doi: 10.1007/s00222-007-0108-4. |
[4] |
T. Clark and S. van Strien, Quasisymmetric rigidity in one-dimensional dynamics, manuscript, 2014. |
[5] |
M. Cortez and J. Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps, Israel. J. Math., 176 (2010), 157-193.
doi: 10.1007/s11856-010-0024-y. |
[6] |
M. Cortez and J. Rivera-Letelier, Choquet simplices as spaces of invariant probability measures on post-critical sets, Ann. Henri Poincaré, 27 (2010), 95-115.
doi: 10.1016/j.anihpc.2009.07.008. |
[7] |
A. Douady, Disques de Siegel et aneaux de Herman, Sém. Bourbaki 1986/87, Astérisque, 1986/87 (1987), 151-172. |
[8] |
G. Estevez and E. de Faria, Real bounds and quasisymmetric rigidity for multicritical circle maps, in preparation. |
[9] |
E. de Faria, Proof of Universality for Critical Circle Mappings, Ph.D. Thesis, CUNY, 1992. |
[10] |
E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergod. Th. & Dynam. Sys., 19 (1999), 995-1035.
doi: 10.1017/S0143385799133959. |
[11] |
E. de Faria and W. de Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc., 1 (1999), 339-392.
doi: 10.1007/s100970050011. |
[12] |
E. de Faria and W. de Melo, Rigidity of critical circle mappings II, J. Amer. Math. Soc., 13 (2000), 343-370.
doi: 10.1090/S0894-0347-99-00324-0. |
[13] |
E. de Faria, W. de Melo and A. Pinto, Global hyperbolicity of renormalization for $C^r$ unimodal mappings, Ann. of Math., 164 (2006), 731-824.
doi: 10.4007/annals.2006.164.731. |
[14] |
P. Guarino, Rigidity Conjecture for $C^3$ Critical Circle Maps, Ph.D. Thesis, IMPA, 2012, available at www.preprint.impa.br. |
[15] |
P. Guarino, M. Martens and W. de Melo, Rigidity of smooth critical circle maps: unbounded combinatorics, in preparation. |
[16] |
P. Guarino and W. de Melo, Rigidity of smooth critical circle maps, available at arXiv:1303.3470. |
[17] |
J. Guckenheimer, Sensitive Dependence to Initial Conditions for One Dimensional Maps, Commun. Math. Phys., 70 (1979), 133-160.
doi: 10.1007/BF01982351. |
[18] |
G. R. Hall, A $C^\infty$ Denjoy counterexample, Ergod. Th. & Dynam. Sys., 1 (1981), 261-272. |
[19] |
M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. |
[20] |
M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations, manuscript, 1988. |
[21] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511809187. |
[22] |
G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergod. Th. & Dynam. Sys., 10 (1990), 717-744.
doi: 10.1017/S0143385700005861. |
[23] |
K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities, Invent. Math., 169 (2007), 193-218.
doi: 10.1007/s00222-007-0047-0. |
[24] |
A. Ya. Khinchin, Continued Fractions, (reprint of the 1964 translation), Dover Publications, Inc., 1997. |
[25] |
D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps, Mosc. Math. J., 6 (2006), 317-351. |
[26] |
S. Lang, Introduction to Diophantine Approximations, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-1-4612-4220-8. |
[27] |
S. Li and W. Shen, Hausdorff dimension of Cantor attractors in one-dimensional dynamics, Invent. Math., 171 (2008), 345-387.
doi: 10.1007/s00222-007-0083-9. |
[28] |
W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-3-642-78043-1. |
[29] |
T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680.
doi: 10.1007/s002220050236. |
[30] |
C. L. Petersen, The Herman-Świątek theorems with applications, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 274 (2000), 211-225. |
[31] |
F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc., 119 (1993), 309-317.
doi: 10.1090/S0002-9939-1993-1186141-9. |
[32] |
F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63.
doi: 10.1007/s00222-002-0243-x. |
[33] |
F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Scient. Éc. Norm. Sup., 40 (2007), 135-178.
doi: 10.1016/j.ansens.2006.11.002. |
[34] |
F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010.
doi: 10.1017/CBO9781139193184. |
[35] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps, available at arXiv:1204.3071. |
[36] |
D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), 417-466, Amer. Math. Soc., Providence, RI, 1992. |
[37] |
G. Świątek, Rational rotation numbers for maps of the circle, Commun. Math. Phys., 119 (1988), 109-128.
doi: 10.1007/BF01218263. |
[38] |
M. Yampolsky, Complex bounds for renormalization of critical circle maps, Ergod. Th. & Dynam. Sys., 19 (1999), 227-257.
doi: 10.1017/S0143385799120947. |
[39] |
M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Commun. Math. Phys., 218 (2001), 537-568.
doi: 10.1007/PL00005561. |
[40] |
M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. IHES, 96 (2002), 1-41.
doi: 10.1007/s10240-003-0007-1. |
[41] |
M. Yampolsky, Renormalization horseshoe for critical circle maps, Commun. Math. Phys., 240 (2003), 75-96.
doi: 10.1007/s00220-003-0891-8. |
[42] |
J.-C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C.R. Acad. Sc. Paris, 298 (1984), 141-144. |
[43] |
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks, Invent. Math., 185 (2011), 421-466.
doi: 10.1007/s00222-011-0312-0. |
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