April  2016, 36(4): 1957-1982. doi: 10.3934/dcds.2016.36.1957

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

Received  July 2014 Revised  July 2015 Published  September 2015

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfies the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.
Citation: Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
References:
[1]

A. Avila, On rigidity of critical circle maps,, Bull. Braz. Math. Soc., 44 (2013), 611.  doi: 10.1007/s00574-013-0027-5.  Google Scholar

[2]

A. Avila, X. Buff and A. Chéritat, Siegel disks with smooth boundaries,, Acta Math., 193 (2004), 1.  doi: 10.1007/BF02392549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Douady, Disques de Siegel et aneaux de Herman,, Sém. Bourbaki 1986/87, 1986/87 (1987), 151.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. R. Hall, A $C^\infty$ Denjoy counterexample,, Ergod. Th. & Dynam. Sys., 1 (1981), 261.   Google Scholar

[19]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. Math. IHES, 49 (1979), 5.   Google Scholar

[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.  Google Scholar

[22]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergod. Th. & Dynam. Sys., 10 (1990), 717.  doi: 10.1017/S0143385700005861.  Google Scholar

[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.  Google Scholar

[24]

A. Ya. Khinchin, Continued Fractions,, (reprint of the 1964 translation), (1964).   Google Scholar

[25]

D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps,, Mosc. Math. J., 6 (2006), 317.   Google Scholar

[26]

S. Lang, Introduction to Diophantine Approximations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4220-8.  Google Scholar

[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.  Google Scholar

[28]

W. de Melo and S. van Strien, One-dimensional Dynamics,, Springer-Verlag, (1995).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[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.  Google Scholar

[30]

C. L. Petersen, The Herman-Świątek theorems with applications,, The Mandelbrot set, 274 (2000), 211.   Google Scholar

[31]

F. Przytycki, Lyapunov characteristic exponents are nonnegative,, Proc. Amer. Math. Soc., 119 (1993), 309.  doi: 10.1090/S0002-9939-1993-1186141-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society Lecture Note Series, 371 (2010).  doi: 10.1017/CBO9781139193184.  Google Scholar

[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.   Google Scholar

[37]

G. Świątek, Rational rotation numbers for maps of the circle,, Commun. Math. Phys., 119 (1988), 109.  doi: 10.1007/BF01218263.  Google Scholar

[38]

M. Yampolsky, Complex bounds for renormalization of critical circle maps,, Ergod. Th. & Dynam. Sys., 19 (1999), 227.  doi: 10.1017/S0143385799120947.  Google Scholar

[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.  Google Scholar

[40]

M. Yampolsky, Hyperbolicity of renormalization of critical circle maps,, Publ. Math. IHES, 96 (2002), 1.  doi: 10.1007/s10240-003-0007-1.  Google Scholar

[41]

M. Yampolsky, Renormalization horseshoe for critical circle maps,, Commun. Math. Phys., 240 (2003), 75.  doi: 10.1007/s00220-003-0891-8.  Google Scholar

[42]

J.-C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique,, C.R. Acad. Sc. Paris, 298 (1984), 141.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Avila, X. Buff and A. Chéritat, Siegel disks with smooth boundaries,, Acta Math., 193 (2004), 1.  doi: 10.1007/BF02392549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Douady, Disques de Siegel et aneaux de Herman,, Sém. Bourbaki 1986/87, 1986/87 (1987), 151.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. R. Hall, A $C^\infty$ Denjoy counterexample,, Ergod. Th. & Dynam. Sys., 1 (1981), 261.   Google Scholar

[19]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Publ. Math. IHES, 49 (1979), 5.   Google Scholar

[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.  Google Scholar

[22]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergod. Th. & Dynam. Sys., 10 (1990), 717.  doi: 10.1017/S0143385700005861.  Google Scholar

[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.  Google Scholar

[24]

A. Ya. Khinchin, Continued Fractions,, (reprint of the 1964 translation), (1964).   Google Scholar

[25]

D. Khmelev and M. Yampolsky, The rigidity problem for analytic critical circle maps,, Mosc. Math. J., 6 (2006), 317.   Google Scholar

[26]

S. Lang, Introduction to Diophantine Approximations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4220-8.  Google Scholar

[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.  Google Scholar

[28]

W. de Melo and S. van Strien, One-dimensional Dynamics,, Springer-Verlag, (1995).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[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.  Google Scholar

[30]

C. L. Petersen, The Herman-Świątek theorems with applications,, The Mandelbrot set, 274 (2000), 211.   Google Scholar

[31]

F. Przytycki, Lyapunov characteristic exponents are nonnegative,, Proc. Amer. Math. Soc., 119 (1993), 309.  doi: 10.1090/S0002-9939-1993-1186141-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society Lecture Note Series, 371 (2010).  doi: 10.1017/CBO9781139193184.  Google Scholar

[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.   Google Scholar

[37]

G. Świątek, Rational rotation numbers for maps of the circle,, Commun. Math. Phys., 119 (1988), 109.  doi: 10.1007/BF01218263.  Google Scholar

[38]

M. Yampolsky, Complex bounds for renormalization of critical circle maps,, Ergod. Th. & Dynam. Sys., 19 (1999), 227.  doi: 10.1017/S0143385799120947.  Google Scholar

[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.  Google Scholar

[40]

M. Yampolsky, Hyperbolicity of renormalization of critical circle maps,, Publ. Math. IHES, 96 (2002), 1.  doi: 10.1007/s10240-003-0007-1.  Google Scholar

[41]

M. Yampolsky, Renormalization horseshoe for critical circle maps,, Commun. Math. Phys., 240 (2003), 75.  doi: 10.1007/s00220-003-0891-8.  Google Scholar

[42]

J.-C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique,, C.R. Acad. Sc. Paris, 298 (1984), 141.   Google Scholar

[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.  Google Scholar

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