Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-08T15:27:12.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic theory for axiom A endomorphisms

Published online by Cambridge University Press:  19 September 2008

Min Qian  and
Zhusheng Zhang
Min Qian
Affiliation:
Department of Mathematics, Peking University, Beijing, China100871
Zhusheng Zhang
Affiliation:
Department of Mathematics, 253-37, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper the Pesin's entropy formula and ‘large ergodic theorem’ are constructed for Axiom A endomorphisms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 161 - 174
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bowen, R.. Some systems with unique equilibrium states. Math. System Theory 8 (3) (1974), 193–202.Google Scholar
[2]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer Lecture Notes in Mathematics Vol. 470. Springer: Berlin, 1975.Google Scholar
[3]Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[4]Brin, M. and Katok, A.. On local entropy. Springer Lecture Notes in Mathematics, Vol. 1007. Springer: Berlin, 30–38.Google Scholar
[5]Zao-Ping, Chen, Lian-Fa, He and Shi-Long, Yang. Orbit shift Ω-stability. Scientia Sinica (Series A) XXXI (1988), 513–594.Google Scholar
[6]Hu-Yi, Hu. Ergodic theorem, Lyapunov characteristic exponents. Adv. Math. 15 (1986), 113–129. 251–267. (In Chinese.)Google Scholar
[7]Hu-Yi, Hu. Pesin's formula for an expanding endomorphism. Adv. Math. (China) 19 (1990), 338–349.Google Scholar
[8]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51, 137–174.Google Scholar
[9]Katok, A. and Strelcyn, Jean-Marie. Invariant manifold, entropy and billiards; smooth maps with singularities. Springer Lecture Notes in Mathematics, Vol. 1222.Google Scholar
[10]Ledrappier, F. and Strelcyn, J.-M.. A proof of the estimation from below in Pensin's entropy formula. Ergod. Th. & Dynam. Sys. 2 (1982), 203–219.CrossRefGoogle Scholar
[11]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. Math. 122 (1985), 509–539. 540–574.CrossRefGoogle Scholar
[12]Mane, R. and Pugh, C.. Stability of endomorphisms. Warwick Dynamical Systems (1974), 175–184.Google Scholar
[13]Pesin, Ia. B.. Lyapunov characteristic exponents and smooth ergodic theory. Russian Math. Surv. 32 (1977), 55–114.Google Scholar
[14]Przytycki, F.. Anosov endomrophisms. Studia Math. 58 (1976), 249–285.Google Scholar
[15]Przytycki, F.. On Ω-stability and structural stability of endomorphisms satisfying Axiom A. Studia Math. 60 (1977), 61–77.Google Scholar
[16]Min, Qian, Zheng-Dong, Wang and Shi-Long, Yang. Structural stability of semi differentiable dynamical systems. Scientica Sinica. Science in China A 32 (1989), 275–282.Google Scholar
[17]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Sot: Bras. Math. 9 (1987), 83–87.Google Scholar
[18]Ruelle, D.. A measure associated with Axiom A attractors. Amer. J. Math. 98 (1976), 619–654.Google Scholar
[19]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: New York, 1978.Google Scholar
[20]Schechter, M.. Principles of Functional Analysis. Academic: New York, 1971.Google Scholar
[21]Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91 (1969), 171–199.CrossRefGoogle Scholar
[22]Sinai, Ya.. Gibbs measures in ergodic theory. Russ. Math. Surv. 166 (1972), 21–69.Google Scholar
[23]Walters, P.. An Introduction to Ergodic Theory. Springer: Berlin, 1982.Google Scholar
[24]Shi-Long, Yang. ‘Many-to-one’ hyperbolic mappings and hyperbolic invariant sets. Acta Math. Sinica 29 (1986), 420–427. (In Chinese.)Google Scholar
[25]Shi-Long, Yang. The orbit shift structural stability of hyperbolic self-covering mappings. Acta Math. Sinica 29 (1986), 590–594. (In Chinese.)Google Scholar
[26]Zhusheng, Zhang. Expanding invariant sets of self-mappings. Scientia Sinica (Series A) XXVII (1984), 1035–1045.Google Scholar
[27]Zhusheng, Zhang. On the shift invariant sets of endomorphisms. Acta Math. Sinica 27 (1984), 564–576. (In Chinese.)Google Scholar
[28]Zhusheng, Zhang. On the Zeta functions of expanding mappings. Scientia Sinica (Series A) XXVIII (1985), 1036–1047.Google Scholar
[29]Zhusheng, Zhang. Principles of Differentiable Dynamical Systems. Science Publisher: China, 1987.Google Scholar