Abstract
We review definitions of random hyperbolic sets and introduce a characterization using random cones. Moreover we discuss problems connected with symbolic representations and the thermodynamic formalism for random hyperbolic systems both in discrete and continuous time cases. In the discrete time case we prove the existence of Markov partitions to guarantee symbolic dynamics and the existence of SRB-measures, while in the continuous time case we explain why a respective method does not work. We illustrate the theory with a number of examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
6 References
L. Arnold. Random Dynamical Systems. Springer, New York, Berlin, 1998.
P. Arnoux and A. Fisher. Anosov families, renormalization and the Teichmüller flow. Preprint, 1995.
P. Baxendale. Asymptotic behaviour of stochastic flows of diffeomorphism. In K. Ito and T. Hida, editors, Stochastic Processes and their Applications. Proc. Nagoya 1985., volume 1203 of Lecture Notes in Math., pages 1–19, New York, Berlin, 1986. Springer.
T. Bogenschütz. Equilibrium States for Random Dynamical Systems. PhD thesis, Universität Bremen, 1993.
T. Bogenschütz and V. M. Gundlach. Symbolic dynamics for expanding random dynamical systems. Random & Computational Dynamics, 1:219–227, 1992.
R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Math. Springer, New York, Berlin, 1975.
R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math, 29:181–202, 1975.
V. M. Gundlach. Random homoclinic orbits. Random & Computational Dynamics, 3(1 & 2):1–33, 1995.
V. M. Gundlach. Thermodynamic formalism for random subshifts of finite type. Report 385, Institut für Dynamische Systeme, Universität Bremen, 1996. revised 1998.
V. M. Gundlach and O. Steinkamp. Products of random rectangular matrices. Report 372, Institut für Dynamische Systeme, Universität Bremen, 1996. To appear in Mathematische Nachrichten.
V. M. Gundlach. Random shifts and matrix products. Habilitationsschrift (in preparation), 1998.
V. M. Gundlach and Y. Kifer. Random sofic systems. Preprint, 1998.
A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press, Cambridge, 1995.
K. Khanin and Y. Kifer. Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. Ser. 2, 171:107–140, 1996.
Y. Kifer. Equilibrium states for random expanding transformations. Random & Computational Dynamics, 1:1–31, 1992.
Y. Kifer. Large deviations for paths and configurations counting. In M. Pollicott and K. Schmidt, editors, Ergodic Theory of \( \mathbb{Z}^d \) -Actions, volume 228 of London Mathematical Society Lecture Note Series, pages 415–432, Cambridge, 1996. Cambridge University Press.
Y. Kifer. Limit theorems for random transformations and processes in random environments. Trans. Amer. Math. Soc., 350:1481–1518, 1998.
P.-D. Liu. Invariant manifolds and the Pesin formula for RDS. In preparation for the Mathematical Physics Series of Springer Verlag’s Encyclopaedia of Mathematical Sciences, 1998.
P.-D. Liu. Random perturbations of Axiom A sets. Jour. Stat. Phys., 90:467–490, 1998.
P.-D. Liu and M. Qian. Smooth Ergodic Theory of Random Dynamical Systems, volume 1606 of Lecture Notes in Math. Springer, New York, Berlin, 1995.
Y. B. Pesin and Y. G. Sinai. Gibbs measures for partially hyperbolic attractors. Ergodic Theory Dynamical Systems, 2:417–438, 1982.
M. Shub. Global Stability of Dynamical Systems. Springer, New York, Berlin, 1987.
O. Steinkamp. Melnikov’s Method and Homoclinic Chaos for Random Dynamical Systems. PhD thesis, Universität Bremen, 1998. In preparation.
T. Wanner. Linearization of random dynamical systems. In C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther, editors, Dynamics Reported, Vol. 4, pages 203–269, New York, Berlin, 1995. Springer.
Rights and permissions
Copyright information
© 1999 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Gundlach, V.M., Kifer, Y. (1999). Random Hyperbolic Systems. In: Stochastic Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-22655-9_6
Download citation
DOI: https://doi.org/10.1007/0-387-22655-9_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98512-1
Online ISBN: 978-0-387-22655-2
eBook Packages: Springer Book Archive