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April  2013, 33(4): 1451-1476. doi: 10.3934/dcds.2013.33.1451

Hyperbolic measures with transverse intersections of stable and unstable manifolds

Michihiro Hirayama 1, and  Naoya Sumi 2,

1. 

Faculty of Engineering, Kyushu Institute of Technology, Tobata, Fukuoka 804-8550, Japan
2. 

Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

Received  January 2011 Revised  September 2012 Published  October 2012

Let $f$ be a diffeomorphism of a manifold preserving a hyperbolic Borel probability measure $ μ $ having transverse intersections for almost every pair of stable and unstable manifolds. A lower bound on the Hausdorff dimension of generic sets is given in terms of the Lyapunov exponents and the metric entropy. Furthermore we obtain a lower bound for the large deviation rate.
Keywords: multifractal analysis., Hyperbolic measures, large deviations.
Mathematics Subject Classification: 37A50, 37C40, 37C45, 37D2.
Citation: Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451
References:
[1]

V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps,, J. Stat. Phys., 125 (2006), 411.  doi: 10.1007/s10955-006-9183-y.  Google Scholar

[2]

L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", Univ. Lect. Ser. 23, (2002).   Google Scholar

[3]

L. Barreira, Ya. B. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures,, Ann. of Math., 149 (1999), 755.  doi: 10.2307/121072.  Google Scholar

[4]

L. R. Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 28 (2008), 587.   Google Scholar

[5]

R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[6]

M. Brin, Hölder continuity of invariant distributions,, in, (2001), 91.   Google Scholar

[7]

M. Brin, J. Feldman and A. B. Katok, Bernoulli diffeomorphisms and group extensions of dynamical systems with non-zero characteristic exponents,, Ann. of. Math., 113 (1981), 159.  doi: 10.2307/1971136.  Google Scholar

[8]

Y. Cao, S. Luzzatto and I. Rios, The boundary of hyperbolicity for Hénon-like families,, Ergod. Th. & Dynam. Sys., 28 (2008), 1049.   Google Scholar

[9]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, in, 66-67 (1979), 66.   Google Scholar

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D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv.Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar

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O. Frostman, Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions,, Meddel. Lunds Univ. Math. Sem., 3 (1935), 1.   Google Scholar

[12]

M. Gerber and A. B. Katok, Smooth models of Thurston's pseudo-Anosov maps,, Ann. scient. Éc. Norm. Sup., 15 (1982), 173.   Google Scholar

[13]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Ètudes Sci. Publ. Math., 51 (1980), 137.  doi: 10.1007/BF02684777.  Google Scholar

[14]

A. B. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).   Google Scholar

[15]

A. B. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior,, Supplement to, (1995), 659.   Google Scholar

[16]

Y. Kifer, Large deviations in dynamical systems and stochastic processes,, Trans. Amer. Math. Soc., 321 (1990), 505.  doi: 10.1090/S0002-9947-1990-1025756-7.  Google Scholar

[17]

F. Ledrappier and J. M. Strelcyn, A proof of estimation from below in Pesin's entropy formula,, Ergod. Th. & Dynam. Sys., 2 (1982), 203.   Google Scholar

[18]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Part I : Characterization of measures satisfying Pesin's entropy formula,, Ann. of Math., 122 (1985), 509.  doi: 10.2307/1971328.  Google Scholar

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Part II: Relations between entropy, exponents and dimension,, Ann. of Math., 122 (1985), 540.  doi: 10.2307/1971329.  Google Scholar

[20]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383.  doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,", Springer, (1987).   Google Scholar

[22]

I. Melbourne and M. Nicol, Large deviations for non-uniformly hyperbolic systems,, Trans. Amer. Math. Soc., 360 (2008), 6661.  doi: 10.1090/S0002-9947-08-04520-0.  Google Scholar

[23]

S. Orey and S. Pelikan, Deviations of trajectory averages and the defect in Pesin's formula for Anosov diffeomorphisms,, Trans. Amer. Math. Soc., 315 (1989), 741.  doi: 10.2307/2001304.  Google Scholar

[24]

Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Izv. Akd. Nauk SSSR Ser. Mat., 40 (1976), 1332.   Google Scholar

[25]

Ya. B. Pesin, "Dimension Theory in Dynamical Systems: Contemporary Views and Applications,", Chicago Lect. Math. Ser., (1997).   Google Scholar

[26]

Ya. B. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions,, J. Stat. Phys., 86 (1997), 233.  doi: 10.1007/BF02180206.  Google Scholar

[27]

Ya. B. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples,, Chaos, 7 (1997), 89.   Google Scholar

[28]

C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta $-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar

[29]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. & Dynam. Sys., 27 (2007), 929.  doi: 10.1017/S0143385706000824.  Google Scholar

[30]

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations,, Ergod. Th. & Dynam. Sys., 26 (2006), 281.  doi: 10.1017/S0143385705000568.  Google Scholar

[31]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property,, Proc. Amer. Math. Soc., 138 (2010), 315.  doi: 10.1090/S0002-9939-09-10085-0.  Google Scholar

[32]

J. Schmeling and H. Weiss, An overview of the dimension theory of dynamical systems,, in, (2001), 429.   Google Scholar

[33]

Y. Takahashi, Two aspects of large deviation theory for large time,, in, (1987), 363.   Google Scholar

[34]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification,, Commun. Math. Phys., 203 (1999), 593.  doi: 10.1007/s002200050627.  Google Scholar

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain noncompact sets,, Ergod. Th. & Dynam. Sys., 23 (2003), 317.   Google Scholar

[36]

M. Urbański and C. Wolf, Ergodic Theory of Parabolic Horseshoes,, Commun. Math. Phys., 281 (2008), 711.  doi: 10.1007/s00220-008-0498-1.  Google Scholar

[37]

L.-S. Young, Dimension, entropy and Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 2 (1982), 109.   Google Scholar

[38]

L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525.  doi: 10.2307/2001318.  Google Scholar

show all references

References:
[1]

V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps,, J. Stat. Phys., 125 (2006), 411.  doi: 10.1007/s10955-006-9183-y.  Google Scholar

[2]

L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", Univ. Lect. Ser. 23, (2002).   Google Scholar

[3]

L. Barreira, Ya. B. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures,, Ann. of Math., 149 (1999), 755.  doi: 10.2307/121072.  Google Scholar

[4]

L. R. Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems,, Ergod. Th. & Dynam. Sys., 28 (2008), 587.   Google Scholar

[5]

R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[6]

M. Brin, Hölder continuity of invariant distributions,, in, (2001), 91.   Google Scholar

[7]

M. Brin, J. Feldman and A. B. Katok, Bernoulli diffeomorphisms and group extensions of dynamical systems with non-zero characteristic exponents,, Ann. of. Math., 113 (1981), 159.  doi: 10.2307/1971136.  Google Scholar

[8]

Y. Cao, S. Luzzatto and I. Rios, The boundary of hyperbolicity for Hénon-like families,, Ergod. Th. & Dynam. Sys., 28 (2008), 1049.   Google Scholar

[9]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces,, in, 66-67 (1979), 66.   Google Scholar

[10]

D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv.Math., 169 (2002), 58.  doi: 10.1006/aima.2001.2054.  Google Scholar

[11]

O. Frostman, Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions,, Meddel. Lunds Univ. Math. Sem., 3 (1935), 1.   Google Scholar

[12]

M. Gerber and A. B. Katok, Smooth models of Thurston's pseudo-Anosov maps,, Ann. scient. Éc. Norm. Sup., 15 (1982), 173.   Google Scholar

[13]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Ètudes Sci. Publ. Math., 51 (1980), 137.  doi: 10.1007/BF02684777.  Google Scholar

[14]

A. B. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995).   Google Scholar

[15]

A. B. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior,, Supplement to, (1995), 659.   Google Scholar

[16]

Y. Kifer, Large deviations in dynamical systems and stochastic processes,, Trans. Amer. Math. Soc., 321 (1990), 505.  doi: 10.1090/S0002-9947-1990-1025756-7.  Google Scholar

[17]

F. Ledrappier and J. M. Strelcyn, A proof of estimation from below in Pesin's entropy formula,, Ergod. Th. & Dynam. Sys., 2 (1982), 203.   Google Scholar

[18]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Part I : Characterization of measures satisfying Pesin's entropy formula,, Ann. of Math., 122 (1985), 509.  doi: 10.2307/1971328.  Google Scholar

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, Part II: Relations between entropy, exponents and dimension,, Ann. of Math., 122 (1985), 540.  doi: 10.2307/1971329.  Google Scholar

[20]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383.  doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,", Springer, (1987).   Google Scholar

[22]

I. Melbourne and M. Nicol, Large deviations for non-uniformly hyperbolic systems,, Trans. Amer. Math. Soc., 360 (2008), 6661.  doi: 10.1090/S0002-9947-08-04520-0.  Google Scholar

[23]

S. Orey and S. Pelikan, Deviations of trajectory averages and the defect in Pesin's formula for Anosov diffeomorphisms,, Trans. Amer. Math. Soc., 315 (1989), 741.  doi: 10.2307/2001304.  Google Scholar

[24]

Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Izv. Akd. Nauk SSSR Ser. Mat., 40 (1976), 1332.   Google Scholar

[25]

Ya. B. Pesin, "Dimension Theory in Dynamical Systems: Contemporary Views and Applications,", Chicago Lect. Math. Ser., (1997).   Google Scholar

[26]

Ya. B. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions,, J. Stat. Phys., 86 (1997), 233.  doi: 10.1007/BF02180206.  Google Scholar

[27]

Ya. B. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples,, Chaos, 7 (1997), 89.   Google Scholar

[28]

C.-E. Pfister and W. G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the $\beta $-shifts,, Nonlinearity, 18 (2005), 237.  doi: 10.1088/0951-7715/18/1/013.  Google Scholar

[29]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets,, Ergod. Th. & Dynam. Sys., 27 (2007), 929.  doi: 10.1017/S0143385706000824.  Google Scholar

[30]

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations,, Ergod. Th. & Dynam. Sys., 26 (2006), 281.  doi: 10.1017/S0143385705000568.  Google Scholar

[31]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property,, Proc. Amer. Math. Soc., 138 (2010), 315.  doi: 10.1090/S0002-9939-09-10085-0.  Google Scholar

[32]

J. Schmeling and H. Weiss, An overview of the dimension theory of dynamical systems,, in, (2001), 429.   Google Scholar

[33]

Y. Takahashi, Two aspects of large deviation theory for large time,, in, (1987), 363.   Google Scholar

[34]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification,, Commun. Math. Phys., 203 (1999), 593.  doi: 10.1007/s002200050627.  Google Scholar

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain noncompact sets,, Ergod. Th. & Dynam. Sys., 23 (2003), 317.   Google Scholar

[36]

M. Urbański and C. Wolf, Ergodic Theory of Parabolic Horseshoes,, Commun. Math. Phys., 281 (2008), 711.  doi: 10.1007/s00220-008-0498-1.  Google Scholar

[37]

L.-S. Young, Dimension, entropy and Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 2 (1982), 109.   Google Scholar

[38]

L.-S. Young, Some large deviation results for dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525.  doi: 10.2307/2001318.  Google Scholar

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