Generalized physical and SRB measures for hyperbolic diffeomorphisms. (English) Zbl 1092.37019
Summary: We introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hénon maps. For this class of systems, we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hénon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs, we apply various techniques from smooth ergodic theory including the thermodynamic formalism.
MSC:
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37C45 | Dimension theory of smooth dynamical systems |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
References:
| [1] | R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470 (Springer-Verlag, Berlin, 1975). · Zbl 0308.28010 |
| [2] | L. Barreira and C. Wolf, Measures of maximal dimension for hyperbolic diffeomorphisms, Comm. Math. Phys. 239:93–113 (2003). · Zbl 1024.37025 · doi:10.1007/s00220-003-0858-9 |
| [3] | L. Barreira and C. Wolf, Local dimension and ergodic decompositions, Ergod. Theory and Dyn. Syst., to appear. |
| [4] | E. Bedford and J. Smillie, Polynomial diffeomorphisms of C 2, Currents, equilibrium measure and hyperbolicity, Invent. Math. 103:69–99 (1991). · Zbl 0721.58037 · doi:10.1007/BF01239509 |
| [5] | E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Annals of Math., to appear. · Zbl 1084.37034 |
| [6] | E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: 2. Small Jacobian, preprint. · Zbl 1084.37034 |
| [7] | M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. 133:73–169 (1991). · Zbl 0724.58042 · doi:10.2307/2944326 |
| [8] | M. Benedicks and L-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math. 112:541–576 (1993). · Zbl 0796.58025 · doi:10.1007/BF01232446 |
| [9] | M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math. 527 (Springer-Verlag, Berlin, 1976). · Zbl 0328.28008 |
| [10] | R. Devaney, and Z. Nitecki, Shift automorphisms in the Hénon family, Comm. Math. Phys. 67:137–148 (1979). · Zbl 0414.58028 · doi:10.1007/BF01221362 |
| [11] | S. Friedland and J. Milnor, Ynamical properties of plane polynomial automorphisms, Ergod. Theory and Dyn. Sys. 9:67–99 (1989). · Zbl 0651.58027 |
| [12] | S. Friedland and G. Ochs, Hausdorff dimension, strong hyperbolicity and complex dynamics, Discrete and Continuous Dyn. Syst. 3:405–430 (1998). · Zbl 0954.37019 |
| [13] | S. Hayes and C. Wolf, Dynamics of a one-parameter family of Hénon maps, Dyn. Sys., to appear. · Zbl 1370.37088 |
| [14] | A. Katok and B. Hasselblatt, Indroduction to modern theory of dynamical systems, (Cambridge University Press, 1995). · Zbl 0878.58020 |
| [15] | R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.) 20:1–24 (1990). · Zbl 0723.58029 · doi:10.1007/BF02585431 |
| [16] | A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergodic Theory Dynamical Systems 1(4):451–459 (1981, 1982). · Zbl 0487.58011 |
| [17] | A. Manning and H. McCluskey, Hausdorff dimension for Horseshoes, Ergod. Theory and Dyn. Syst. 3:251–260 (1983). · Zbl 0529.58022 |
| [18] | M. Rams, Measures of maximal dimension for linear horseshoes, preprint. · Zbl 1096.37013 |
| [19] | Ya. Pesin, Dimension theory in dynamical systems: Contemporary views and applications, Chicago Lectures in Mathematics (Chicago University Press, 1997). |
| [20] | A. Pinto and D. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc. 34(3):341–352 (2002). · Zbl 1027.37016 · doi:10.1112/S0024609301008670 |
| [21] | D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, MA, 1978). |
| [22] | D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187(1):227–241 (1997). · Zbl 0895.58045 · doi:10.1007/s002200050134 |
| [23] | R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension, Discrete and Continuous Dyn. Syst. 12:403–412 (2005). · Zbl 1069.37018 |
| [24] | C. Wolf, Dimension of Julia sets of polynomial automorphisms of C 2, Michigan Mathematical Journal 47:585–600 (2000). · Zbl 1053.37025 · doi:10.1307/mmj/1030132596 |
| [25] | C. Wolf, On measures of maximal and full dimension for polynomial automorphisms of C 2, Trans. Amer. Math. Soc. 355:3227–3239 (2003). · Zbl 1029.37026 · doi:10.1090/S0002-9947-03-03287-2 |
| [26] | L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Theory and Dyn. Syst. 2:109–124 (1982). · Zbl 0523.58024 · doi:10.1017/S0143385700009615 |
| [27] | L.-S. Young, Ergodic theory of differentiable dynamical systems, in Real and Complex Dynamics, Ed. Branner and Hjorth, Nato ASI series, (Kluwer Academic Publishers, 1995) pp. 293–336. · Zbl 0830.58020 |
| [28] | L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys. 108:733–754 (2002). · Zbl 1124.37307 · doi:10.1023/A:1019762724717 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
