Hyperbolic measures with transverse intersections of stable and unstable manifolds

Michihiro Hirayama; Naoya Sumi

doi:10.3934/dcds.2013.33.1451

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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

Abstract
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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
[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

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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Hyperbolic measures with transverse intersections of stable and unstable manifolds

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