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Generic points of shift-invariant measures in the countable symbolic space

Published online by Cambridge University Press:  21 February 2018

AI–HUA FAN
Affiliation:
School of Mathematics and Statistics, Central China Normal University, 152, Luoyu Road, 430079 Wuhan, China LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France. e-mail: [email protected]
MING–TIAN LI
Affiliation:
College of Mathematics and Informatics, Fujian Normal University, Keji Road 350017 Fuzhou, China. e-mail: [email protected]
JI–HUA MA
Affiliation:
Department of Mathematics, Wuhan University, 299 Bayi Road, 430072 Wuhan, China. e-mail: [email protected]

Abstract

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Adler, R. F-expansions revisited. Recent advances in topological dynamics (Proc. Conf., Yale University, New Haven, Connecticut, 1972; in honor of Gustav Arnold Nedlund). Lecture Notes in Math., vol. 318 (Springer, Berlin, 1973), pp. 1-5.Google Scholar
[2] Billingsley, P. Convergence of probability measures (2th edition) (John Wiley & Sons Inc, 1999).Google Scholar
[3] Bowen, R. Topological entropy for noncompact set. Trans. Amer. Math. Soc. 184 (1973), 125-136.Google Scholar
[4] Cajar, H. Billingsley Dimension in Probability Spaces (Springer-Verlag, 1981).Google Scholar
[5] Denker, M., Grillenberger, C. and Sigmund, K. Ergodic theory on compact spaces. Lecture Notes in Math., vol. 527 (Springer-Verlag, Berlin-New York, 1976).Google Scholar
[6] Fan, A. H. and Feng, D. J. On the distribution of long-term time averages on symbolic space. J. Statist. Phys. 99 (2000), no. 3–4, 813-856.Google Scholar
[7] Fan, A. H., Feng, D. J. and Wu, J. Recurence, dimension and entropy. J. London Math. Soc. (2) 64 (2001), 229-244.Google Scholar
[8] Fan, A. H., Liao, L. M. and Ma, J. H. On the frequency of partial quotients of regular continued fractions. Math. Proc. Camb. Phil. Soc. 148 (2010), 179-192.Google Scholar
[9] Fan, A. H., Liao, L. M., Ma, J. H. and Wang, B. W. Dimension of Besicovitch–Eggleston sets in countable symbolic space. Nonlinearity 23 (2010), 1185-1197.Google Scholar
[10] Fan, A. H., Liao, L. M. and Peyriere, J. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 1103-1128.Google Scholar
[11] Fan, A. H., Jordan, T., Liao, L. M. and Rams, M. Multifractal analysis for expanding interval maps with infinitely many branches. Trans. Amer. Math. Soc. 367 (2015), no. 3, 1847-1870.Google Scholar
[12] Glasner, E. and Weiss, B. On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems vol.1B (Elsevier B.V., Amsterdam, 2006), 597-648.Google Scholar
[13] Good, I. J. The fractional dimensional theory of continued fractions. Proc. Camb. Phil. Soc. 37 (1941), 199-228.Google Scholar
[14] Gurevich, B. M. and Tempelman, A. A. Hausdorff dimension of sets of generic points for Gibbs measures. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys. 108 (2002), no. 5–6, 1281-1301.Google Scholar
[15] Liao, L. M., Ma, J. H. and Wang, B. W. Dimension of some non-normal continued fraction sets. Math. Proc. Camb. Phil. Soc. 145 (2008), no. 1, 215-225.Google Scholar
[16] Ma, J. H. and Wen, Z. Y. Hausdorff and packing measure of sets of generic points: a zero-infinity law. J. London Math. Soc. (2) 69 (2004), 383-406.Google Scholar
[17] Mattila, P. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Camb. Stud. Adv. Math. 44 (Cambridge University Press, Cambridge, 1995).Google Scholar
[18] Olivier, E. Dimension de Billingsley d'ensembles saturés. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 1, 13-16.Google Scholar
[19] Parthasarathy, K. R. Probability Measure on Metric Space (Academic Press, New York and London, 1967).Google Scholar
[20] Pfister, C.-E. and Sullivan, W. G. Billingsley dimension on shift spaces. Nonlinearity 16 (2003), no. 2, 661-682.Google Scholar
[21] Pfister, C.-E. and Sullivan, W. G. On the topological entropy of saturated sets. Ergodic Theory Dynam. Systems 27 (2007), no. 3, 929-956.Google Scholar
[22] Pollicott, M. and Weiss, H. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys. 207 (1999), no. 1, 145-171.Google Scholar
[23] Polya, G. and Szego, G. Problems and Theorems in Analysis. Vol. I: Series, Integral Calculus, Theory of Functions (Springer, 1972).Google Scholar
[24] Sarig, O. Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565-1593.Google Scholar
[25] Sarig, O. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751-1558.Google Scholar
[26] Sarig, O. Lecture notes on thermodynamic formalism for topological markov shifts (2009). See also http://www.wisdom.weizmann.ac.il/sarigo/TDFnotes.pdf.Google Scholar
[27] Schweiger, F. Ergodic Theory of Fibred Systems and Metric Number Theory (Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995).Google Scholar
[28] Seneta, E. Non-negative Matrices and Markov Chains (Springer, 2006).Google Scholar
[29] Walters, P. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121-153.Google Scholar
[30] Walters, P. An Introduction to Ergodic Theory (Springer-Verlag, 1982).Google Scholar
[31] Wegmann, H. Über den dimensionsbegriff in wahrscheinlichkeitsrumen, II. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 9 (1968), 222-231.Google Scholar