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Global solutions to a one-dimensional non-conservative two-phase model
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.
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.
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.
doi: 10.1007/s00222-007-0108-4. |
[4] |
T. Clark and S. van Strien, Quasisymmetric rigidity in one-dimensional dynamics,, manuscript, (2014). Google Scholar |
[5] |
M. Cortez and J. Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps,, Israel. J. Math., 176 (2010), 157.
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.
doi: 10.1016/j.anihpc.2009.07.008. |
[7] |
A. Douady, Disques de Siegel et aneaux de Herman,, Sém. Bourbaki 1986/87, 1986/87 (1987), 151.
|
[8] |
G. Estevez and E. de Faria, Real bounds and quasisymmetric rigidity for multicritical circle maps,, in preparation., (). Google Scholar |
[9] |
E. de Faria, Proof of Universality for Critical Circle Mappings,, Ph.D. Thesis, (1992). Google Scholar |
[10] |
E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings,, Ergod. Th. & Dynam. Sys., 19 (1999), 995.
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.
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.
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.
doi: 10.4007/annals.2006.164.731. |
[14] |
P. Guarino, Rigidity Conjecture for $C^3$ Critical Circle Maps,, Ph.D. Thesis, (2012). Google Scholar |
[15] |
P. Guarino, M. Martens and W. de Melo, Rigidity of smooth critical circle maps: unbounded combinatorics,, in preparation., (). Google Scholar |
[16] |
P. Guarino and W. de Melo, Rigidity of smooth critical circle maps,, available at arXiv:1303.3470., (). Google Scholar |
[17] |
J. Guckenheimer, Sensitive Dependence to Initial Conditions for One Dimensional Maps,, Commun. Math. Phys., 70 (1979), 133.
doi: 10.1007/BF01982351. |
[18] |
G. R. Hall, A $C^\infty$ Denjoy counterexample,, Ergod. Th. & Dynam. Sys., 1 (1981), 261.
|
[19] |
M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. Math. IHES, 49 (1979), 5.
|
[20] |
M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations,, manuscript, (1988). Google Scholar |
[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.
doi: 10.1017/S0143385700005861. |
[23] |
K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities,, Invent. Math., 169 (2007), 193.
doi: 10.1007/s00222-007-0047-0. |
[24] |
A. Ya. Khinchin, Continued Fractions,, (reprint of the 1964 translation), (1964).
|
[25] |
D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps,, Mosc. Math. J., 6 (2006), 317.
|
[26] |
S. Lang, Introduction to Diophantine Approximations,, Springer-Verlag, (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.
doi: 10.1007/s00222-007-0083-9. |
[28] |
W. de Melo and S. van Strien, One-dimensional Dynamics,, Springer-Verlag, (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.
doi: 10.1007/s002220050236. |
[30] |
C. L. Petersen, The Herman-Świątek theorems with applications,, The Mandelbrot set, 274 (2000), 211.
|
[31] |
F. Przytycki, Lyapunov characteristic exponents are nonnegative,, Proc. Amer. Math. Soc., 119 (1993), 309.
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.
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.
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 (2010).
doi: 10.1017/CBO9781139193184. |
[35] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, available at arXiv:1204.3071., (). Google Scholar |
[36] |
D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures,, American Mathematical Society centennial publications, (1988), 417.
|
[37] |
G. Świątek, Rational rotation numbers for maps of the circle,, Commun. Math. Phys., 119 (1988), 109.
doi: 10.1007/BF01218263. |
[38] |
M. Yampolsky, Complex bounds for renormalization of critical circle maps,, Ergod. Th. & Dynam. Sys., 19 (1999), 227.
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.
doi: 10.1007/PL00005561. |
[40] |
M. Yampolsky, Hyperbolicity of renormalization of critical circle maps,, Publ. Math. IHES, 96 (2002), 1.
doi: 10.1007/s10240-003-0007-1. |
[41] |
M. Yampolsky, Renormalization horseshoe for critical circle maps,, Commun. Math. Phys., 240 (2003), 75.
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.
|
[43] |
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks,, Invent. Math., 185 (2011), 421.
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.
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.
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.
doi: 10.1007/s00222-007-0108-4. |
[4] |
T. Clark and S. van Strien, Quasisymmetric rigidity in one-dimensional dynamics,, manuscript, (2014). Google Scholar |
[5] |
M. Cortez and J. Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps,, Israel. J. Math., 176 (2010), 157.
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.
doi: 10.1016/j.anihpc.2009.07.008. |
[7] |
A. Douady, Disques de Siegel et aneaux de Herman,, Sém. Bourbaki 1986/87, 1986/87 (1987), 151.
|
[8] |
G. Estevez and E. de Faria, Real bounds and quasisymmetric rigidity for multicritical circle maps,, in preparation., (). Google Scholar |
[9] |
E. de Faria, Proof of Universality for Critical Circle Mappings,, Ph.D. Thesis, (1992). Google Scholar |
[10] |
E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings,, Ergod. Th. & Dynam. Sys., 19 (1999), 995.
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.
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.
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.
doi: 10.4007/annals.2006.164.731. |
[14] |
P. Guarino, Rigidity Conjecture for $C^3$ Critical Circle Maps,, Ph.D. Thesis, (2012). Google Scholar |
[15] |
P. Guarino, M. Martens and W. de Melo, Rigidity of smooth critical circle maps: unbounded combinatorics,, in preparation., (). Google Scholar |
[16] |
P. Guarino and W. de Melo, Rigidity of smooth critical circle maps,, available at arXiv:1303.3470., (). Google Scholar |
[17] |
J. Guckenheimer, Sensitive Dependence to Initial Conditions for One Dimensional Maps,, Commun. Math. Phys., 70 (1979), 133.
doi: 10.1007/BF01982351. |
[18] |
G. R. Hall, A $C^\infty$ Denjoy counterexample,, Ergod. Th. & Dynam. Sys., 1 (1981), 261.
|
[19] |
M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. Math. IHES, 49 (1979), 5.
|
[20] |
M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations,, manuscript, (1988). Google Scholar |
[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.
doi: 10.1017/S0143385700005861. |
[23] |
K. Khanin and A. Teplinsky, Robust rigidity for circle diffeomorphisms with singularities,, Invent. Math., 169 (2007), 193.
doi: 10.1007/s00222-007-0047-0. |
[24] |
A. Ya. Khinchin, Continued Fractions,, (reprint of the 1964 translation), (1964).
|
[25] |
D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps,, Mosc. Math. J., 6 (2006), 317.
|
[26] |
S. Lang, Introduction to Diophantine Approximations,, Springer-Verlag, (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.
doi: 10.1007/s00222-007-0083-9. |
[28] |
W. de Melo and S. van Strien, One-dimensional Dynamics,, Springer-Verlag, (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.
doi: 10.1007/s002220050236. |
[30] |
C. L. Petersen, The Herman-Świątek theorems with applications,, The Mandelbrot set, 274 (2000), 211.
|
[31] |
F. Przytycki, Lyapunov characteristic exponents are nonnegative,, Proc. Amer. Math. Soc., 119 (1993), 309.
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.
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.
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 (2010).
doi: 10.1017/CBO9781139193184. |
[35] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, available at arXiv:1204.3071., (). Google Scholar |
[36] |
D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures,, American Mathematical Society centennial publications, (1988), 417.
|
[37] |
G. Świątek, Rational rotation numbers for maps of the circle,, Commun. Math. Phys., 119 (1988), 109.
doi: 10.1007/BF01218263. |
[38] |
M. Yampolsky, Complex bounds for renormalization of critical circle maps,, Ergod. Th. & Dynam. Sys., 19 (1999), 227.
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.
doi: 10.1007/PL00005561. |
[40] |
M. Yampolsky, Hyperbolicity of renormalization of critical circle maps,, Publ. Math. IHES, 96 (2002), 1.
doi: 10.1007/s10240-003-0007-1. |
[41] |
M. Yampolsky, Renormalization horseshoe for critical circle maps,, Commun. Math. Phys., 240 (2003), 75.
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.
|
[43] |
G. Zhang, All bounded type Siegel disks of rational maps are quasi-disks,, Invent. Math., 185 (2011), 421.
doi: 10.1007/s00222-011-0312-0. |
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