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.

[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).

[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., ().

[9]

E. de Faria, Proof of Universality for Critical Circle Mappings,, Ph.D. Thesis, (1992).

[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).

[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. 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).

[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., ().

[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).

[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., ().

[9]

E. de Faria, Proof of Universality for Critical Circle Mappings,, Ph.D. Thesis, (1992).

[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).

[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. 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).

[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., ().

[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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