Abstract
In this paper 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 Hènon 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 Hènon 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.
Similar content being viewed by others
References
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470 (Springer-Verlag, Berlin, 1975).
L. Barreira and C. Wolf, Measures of maximal dimension for hyperbolic diffeomorphisms, Comm. Math. Phys. 239:93–113 (2003).
L. Barreira and C. Wolf, Local dimension and ergodic decompositions, Ergod. Theory and Dyn. Syst., to appear.
E. Bedford and J. Smillie, Polynomial diffeomorphisms of C 2, Currents, equilibrium measure and hyperbolicity, Invent. Math. 103:69–99 (1991).
E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Annals of Math., to appear.
E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: 2. Small Jacobian, preprint.
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. 133:73–169 (1991).
M. Benedicks and L-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math. 112:541–576 (1993).
M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math. 527 (Springer-Verlag, Berlin, 1976).
R. Devaney, and Z. Nitecki, Shift automorphisms in the Hénon family, Comm. Math. Phys. 67:137–148 (1979).
S. Friedland and J. Milnor, Ynamical properties of plane polynomial automorphisms, Ergod. Theory and Dyn. Sys. 9:67–99 (1989).
S. Friedland and G. Ochs, Hausdorff dimension, strong hyperbolicity and complex dynamics, Discrete and Continuous Dyn. Syst. 3:405–430 (1998).
S. Hayes and C. Wolf, Dynamics of a one-parameter family of Hénon maps, Dyn. Sys., to appear.
A. Katok and B. Hasselblatt, Indroduction to modern theory of dynamical systems, (Cambridge University Press, 1995).
R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.) 20:1–24 (1990).
A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergodic Theory Dynamical Systems 1(4):451–459 (1981, 1982).
A. Manning and H. McCluskey, Hausdorff dimension for Horseshoes, Ergod. Theory and Dyn. Syst. 3:251–260 (1983).
M. Rams, Measures of maximal dimension for linear horseshoes, preprint.
Ya. Pesin, Dimension theory in dynamical systems: Contemporary views and applications, Chicago Lectures in Mathematics (Chicago University Press, 1997).
A. Pinto and D. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc. 34(3):341–352 (2002).
D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, MA, 1978).
D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187(1):227–241 (1997).
R. Shafikov and C. Wolf, Stable sets, hyperbolicity and dimension, Discrete and Continuous Dyn. Syst. 12:403–412 (2005).
C. Wolf, Dimension of Julia sets of polynomial automorphisms of C 2, Michigan Mathematical Journal 47:585–600 (2000).
C. Wolf, On measures of maximal and full dimension for polynomial automorphisms of C 2, Trans. Amer. Math. Soc. 355:3227–3239 (2003).
L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Theory and Dyn. Syst. 2:109–124 (1982).
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.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30
Rights and permissions
About this article
Cite this article
Wolf, C. Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms. J Stat Phys 122, 1111–1138 (2006). https://doi.org/10.1007/s10955-005-8024-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-8024-8