On the weaker forms of the specification property and their applications
HTML articles powered by AMS MathViewer
- by Kenichiro Yamamoto PDF
- Proc. Amer. Math. Soc. 137 (2009), 3807-3814 Request permission
Abstract:
We show the following two results, which are derived from the weaker forms of the specification property: Firstly, if an automorphism of a compact metric abelian group with finite topological entropy is ergodic under the Haar measure, then it satisfies the level 2 large deviation principle. Secondly, the topological pressure formula for periodic orbits is given under the expansiveness and the almost product property.References
- Nobuo Aoki, Zero-dimensional automorphisms with specification, Monatsh. Math. 95 (1983), no. 1, 1–17. MR 697344, DOI 10.1007/BF01301143
- N. Aoki, M. Dateyama, and M. Komuro, Solenoidal automorphisms with specifications, Monatsh. Math. 93 (1982), no. 2, 79–110. MR 653100, DOI 10.1007/BF01301397
- Viviane Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1793194, DOI 10.1142/9789812813633
- Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 399413, DOI 10.1007/BF01762666
- Jérôme Buzzi, Specification on the interval, Trans. Amer. Math. Soc. 349 (1997), no. 7, 2737–2754. MR 1407484, DOI 10.1090/S0002-9947-97-01873-4
- Yong Moo Chung and Michihiro Hirayama, Topological entropy and periodic orbits of saddle type for surface diffeomorphisms, Hiroshima Math. J. 33 (2003), no. 2, 189–195. MR 1997693
- Masahito Dateyama, The almost weak specification property for ergodic group automorphisms of abelian groups, J. Math. Soc. Japan 42 (1990), no. 2, 341–351. MR 1041229, DOI 10.2969/jmsj/04220341
- Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
- Richard S. Ellis, Entropy, large deviations, and statistical mechanics, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 271, Springer-Verlag, New York, 1985. MR 793553, DOI 10.1007/978-1-4613-8533-2
- Katrin Gelfert and Christian Wolf, Topological pressure via saddle points, Trans. Amer. Math. Soc. 360 (2008), no. 1, 545–561. MR 2342015, DOI 10.1090/S0002-9947-07-04407-8
- Lianfa He, Hong Li, and Winxiang Sun, Invariant of topological pressure under some semi-conjugates, Appl. Math. J. Chinese Univ. Ser. B 12 (1997), no. 3, 355–362. A Chinese summary appears in Gaoxiao Yingyong Shuxue Xuebao Ser. A 12 (1997), no. 3, 378. MR 1482923, DOI 10.1007/s11766-997-0036-5
- D. A. Lind, Ergodic group automorphisms and specification, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 93–104. MR 550414
- Brian Marcus, A note on periodic points for ergodic toral automorphisms, Monatsh. Math. 89 (1980), no. 2, 121–129. MR 572888, DOI 10.1007/BF01476590
- MichałMisiurewicz, Topological conditional entropy, Studia Math. 55 (1976), no. 2, 175–200. MR 415587, DOI 10.4064/sm-55-2-175-200
- 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), no. 1, 237–261. MR 2109476, DOI 10.1088/0951-7715/18/1/013
- C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 929–956. MR 2322186, DOI 10.1017/S0143385706000824
- Jörg Schmeling, Symbolic dynamics for $\beta$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 675–694. MR 1452189, DOI 10.1017/S0143385797079182
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Additional Information
- Kenichiro Yamamoto
- Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
- MR Author ID: 878580
- Email: yamamoto.k.ak@m.titech.ac.jp
- Received by editor(s): December 25, 2008
- Received by editor(s) in revised form: February 23, 2009
- Published electronically: June 10, 2009
- Communicated by: Jane M. Hawkins
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3807-3814
- MSC (2000): Primary 37B40; Secondary 60F10
- DOI: https://doi.org/10.1090/S0002-9939-09-09937-7
- MathSciNet review: 2529890