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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 and Continuous Dynamical Systems, 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), 411-453. doi: 10.1007/s10955-006-9183-y.

[2]

L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," Univ. Lect. Ser. 23, Amer. Math. Soc., 2002.

[3]

L. Barreira, Ya. B. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math., 149 (1999), 755-783. doi: 10.2307/121072.

[4]

L. R. Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 28 (2008), 587-612.

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[6]

M. Brin, Hölder continuity of invariant distributions, in "Smooth Ergodic Theory and Its Applications" (eds. A. B. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Proc. Symp. Pure Math., Amer. Math. Soc., (2001), 91-93.

[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), 159-179. doi: 10.2307/1971136.

[8]

Y. Cao, S. Luzzatto and I. Rios, The boundary of hyperbolicity for Hénon-like families, Ergod. Th. & Dynam. Sys., 28 (2008), 1049-1080.

[9]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, in "Séminaire Orsay, Astérisque," 66-67 (1979).

[10]

D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv.Math., 169 (2002), 58-91. doi: 10.1006/aima.2001.2054.

[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), 1-118.

[12]

M. Gerber and A. B. Katok, Smooth models of Thurston's pseudo-Anosov maps, Ann. scient. Éc. Norm. Sup., 15 (1982), 173-204.

[13]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Ètudes Sci. Publ. Math., 51 (1980), 137-173. doi: 10.1007/BF02684777.

[14]

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

[15]

A. B. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior, Supplement to "Introduction to the Modern Theory of Dynamical Systems" (eds. A. B. Katok and B. Hasselblatt), Cambridge University Press, Cambridge, (1995), 659-700.

[16]

Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc., 321 (1990), 505-524. doi: 10.1090/S0002-9947-1990-1025756-7.

[17]

F. Ledrappier and J. M. Strelcyn, A proof of estimation from below in Pesin's entropy formula, Ergod. Th. & Dynam. Sys., 2 (1982), 203-219.

[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), 509-539. doi: 10.2307/1971328.

[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), 540-574. doi: 10.2307/1971329.

[20]

R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer, Berlin, 1987.

[22]

I. Melbourne and M. Nicol, Large deviations for non-uniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0.

[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), 741-753. doi: 10.2307/2001304.

[24]

Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Izv. Akd. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379; English transl., Math. USSR. Isvestia, 40 (1976), 1261-1305.

[25]

Ya. B. Pesin, "Dimension Theory in Dynamical Systems: Contemporary Views and Applications," Chicago Lect. Math. Ser., University of Chicago Press, 1997.

[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), 233-275. doi: 10.1007/BF02180206.

[27]

Ya. B. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106.

[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), 237-261. doi: 10.1088/0951-7715/18/1/013.

[29]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. & Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.

[30]

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations, Ergod. Th. & Dynam. Sys., 26 (2006), 281-289. doi: 10.1017/S0143385705000568.

[31]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc., 138 (2010), 315-321. doi: 10.1090/S0002-9939-09-10085-0.

[32]

J. Schmeling and H. Weiss, An overview of the dimension theory of dynamical systems, in "Smooth Ergodic Theory and Its Applications" (eds. A. B. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Proc. Symp. Pure Math., Amer. Math. Soc., (2001), 429-488.

[33]

Y. Takahashi, Two aspects of large deviation theory for large time, in "Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985)" (eds. A. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Academic Press, Boston, (1987), 363-384

[34]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Commun. Math. Phys., 203 (1999), 593-612. doi: 10.1007/s002200050627.

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain noncompact sets, Ergod. Th. & Dynam. Sys., 23 (2003), 317-348.

[36]

M. Urbański and C. Wolf, Ergodic Theory of Parabolic Horseshoes, Commun. Math. Phys., 281 (2008), 711-751. doi: 10.1007/s00220-008-0498-1.

[37]

L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 2 (1982), 109-124.

[38]

L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.

show all references

References:
[1]

V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps, J. Stat. Phys., 125 (2006), 411-453. doi: 10.1007/s10955-006-9183-y.

[2]

L. Barreira and Ya. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," Univ. Lect. Ser. 23, Amer. Math. Soc., 2002.

[3]

L. Barreira, Ya. B. Pesin and J. Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math., 149 (1999), 755-783. doi: 10.2307/121072.

[4]

L. R. Bellet and L.-S. Young, Large deviations in non-uniformly hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., 28 (2008), 587-612.

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[6]

M. Brin, Hölder continuity of invariant distributions, in "Smooth Ergodic Theory and Its Applications" (eds. A. B. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Proc. Symp. Pure Math., Amer. Math. Soc., (2001), 91-93.

[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), 159-179. doi: 10.2307/1971136.

[8]

Y. Cao, S. Luzzatto and I. Rios, The boundary of hyperbolicity for Hénon-like families, Ergod. Th. & Dynam. Sys., 28 (2008), 1049-1080.

[9]

A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces, in "Séminaire Orsay, Astérisque," 66-67 (1979).

[10]

D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv.Math., 169 (2002), 58-91. doi: 10.1006/aima.2001.2054.

[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), 1-118.

[12]

M. Gerber and A. B. Katok, Smooth models of Thurston's pseudo-Anosov maps, Ann. scient. Éc. Norm. Sup., 15 (1982), 173-204.

[13]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Ètudes Sci. Publ. Math., 51 (1980), 137-173. doi: 10.1007/BF02684777.

[14]

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

[15]

A. B. Katok and L. Mendoza, Dynamical systems with nonuniformly hyperbolic behavior, Supplement to "Introduction to the Modern Theory of Dynamical Systems" (eds. A. B. Katok and B. Hasselblatt), Cambridge University Press, Cambridge, (1995), 659-700.

[16]

Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc., 321 (1990), 505-524. doi: 10.1090/S0002-9947-1990-1025756-7.

[17]

F. Ledrappier and J. M. Strelcyn, A proof of estimation from below in Pesin's entropy formula, Ergod. Th. & Dynam. Sys., 2 (1982), 203-219.

[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), 509-539. doi: 10.2307/1971328.

[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), 540-574. doi: 10.2307/1971329.

[20]

R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8.

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer, Berlin, 1987.

[22]

I. Melbourne and M. Nicol, Large deviations for non-uniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0.

[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), 741-753. doi: 10.2307/2001304.

[24]

Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Izv. Akd. Nauk SSSR Ser. Mat., 40 (1976), 1332-1379; English transl., Math. USSR. Isvestia, 40 (1976), 1261-1305.

[25]

Ya. B. Pesin, "Dimension Theory in Dynamical Systems: Contemporary Views and Applications," Chicago Lect. Math. Ser., University of Chicago Press, 1997.

[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), 233-275. doi: 10.1007/BF02180206.

[27]

Ya. B. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106.

[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), 237-261. doi: 10.1088/0951-7715/18/1/013.

[29]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. & Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.

[30]

E. Pujals and M. Sambarino, A sufficient condition for robustly minimal foliations, Ergod. Th. & Dynam. Sys., 26 (2006), 281-289. doi: 10.1017/S0143385705000568.

[31]

K. Sakai, N. Sumi and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc., 138 (2010), 315-321. doi: 10.1090/S0002-9939-09-10085-0.

[32]

J. Schmeling and H. Weiss, An overview of the dimension theory of dynamical systems, in "Smooth Ergodic Theory and Its Applications" (eds. A. B. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Proc. Symp. Pure Math., Amer. Math. Soc., (2001), 429-488.

[33]

Y. Takahashi, Two aspects of large deviation theory for large time, in "Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985)" (eds. A. Katok, R. de la Llave, Ya. B. Pesin and H. Weiss), Academic Press, Boston, (1987), 363-384

[34]

F. Takens and E. Verbitskiy, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Commun. Math. Phys., 203 (1999), 593-612. doi: 10.1007/s002200050627.

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain noncompact sets, Ergod. Th. & Dynam. Sys., 23 (2003), 317-348.

[36]

M. Urbański and C. Wolf, Ergodic Theory of Parabolic Horseshoes, Commun. Math. Phys., 281 (2008), 711-751. doi: 10.1007/s00220-008-0498-1.

[37]

L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 2 (1982), 109-124.

[38]

L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318.

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