Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Statistical properties for nonhyperbolic maps with finite range structure


Author: Michiko Yuri
Journal: Trans. Amer. Math. Soc. 352 (2000), 2369-2388
MSC (2000): Primary 11K50, 11K55, 28D05, 58F03, 58F11, 58F15
DOI: https://doi.org/10.1090/S0002-9947-00-02579-4
Published electronically: February 14, 2000
MathSciNet review: 1695039
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of $ L^1 $-convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.


References [Enhancements On Off] (What's this?)

  • [1] J. Aaronson. The asymptotic distribution behavior of transformations preserving infinite measures. J. Analyse Math. 39 (1981), 203-234. MR 82m:28030
  • [2] J. Aaronson. Random $ f $-expansions. The Annals of Probability 14 (1986), 1037-1057. MR 87k:60057
  • [3] J. Aaronson. An Introduction to Infinite Ergodic Theory (1997), Amer. Math. Soc. MR 99k:28025
  • [4] J. Aaronson and M. Denker. Lower bounds for partial sums of certain positive stationary process, almost everywhere convergence, Proceedings (G.A. Edgar and L. Sucheston, eds.), Academic Press, (1989) 1-9. MR 91e:28012
  • [5] J. Aaronson and M. Denker. Local limit theorems for Gibbs Markov maps. Preprint.
  • [6] J. Aaronson, M. Denker and A. Fisher. Second order ergodic theorems for ergodic transformations of infinite measure spaces, Proc. Amer. Math. Soc. 114 (1992), 115-127. MR 92e:28007
  • [7] J. Aaronson, M. Denker and M. Urbanski. Ergodic Theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495-548. MR 94g:58116
  • [8] R. Bradley. On the $ {\psi}$-mixing condition for stationary random sequences. Trans. Amer. Math. Soc. 276 (1983), 55-66. MR 84h:60071
  • [9] A. Broise. Fractions continues multidimensionelles et lois stables. Bull. Soc. Math. France 124 (1996), 97-139 MR 97h:11081
  • [10] N. I. Chernov. Limit Theorems and Markov approximations for certain chaotic dynamical systems. Prob. Theor. Relat. Fields. 101 (1995), 321-362. MR 96m:28016
  • [11] D. A. Darling and M. Kac. On occupation times for Markov processes. Trans. Amer. Math. Soc. 84 (1957), 444-458. MR 18:832a
  • [12] R. A. Davis. Stable limits for partial sums of dependent random variables. Ann. Probab. 11 (1983), 262-269. MR 84e:60028
  • [13] P. Gaspard and X. J. Wang. Sporadicity: between periodic and chaotic dynamical behaviours. Proc. Natl. Acad. Sci. USA 85 (1988) 4591-4595. MR 90f:58115
  • [14] M. Gordin. Exponentially fast mixing. Dokl. Akad. Nauk SSSR 96 (1971), 1255-1258. MR 43:2752
  • [15] Ionescu-Tulcea and G. Marinescu. Theorie ergodique pour des classes d'operations non completement continuous. Ann. Math. 47 (1946), 140-147. MR 12:226
  • [16] I. A. Ibragimov and Y. V. Linnik. Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen, Netherlands. (1971). MR 48:1287
  • [17] Sh. Ito and M. Yuri. Number theoretical transformations and their ergodic properties. Tokyo J. Math. 10 (1987) 1-32. MR 88i:28031
  • [18] C. Liverani. Flows, random perturbations and rate of mixing. Ergodic Theory and Dynamical Systems 18 (1998), 1421-1446. CMP 99:05
  • [19] C. Liverani, B. Saussall, and S. Vaianti. A probabilistic approach to Intermittency. Preprint.
  • [20] D. Mayer. Approach to equilibrium for locally expanding maps in $\mathbf{R}^k. $ Commun. Math. Phys. 95 (1984), 1-15. MR 86d:58069
  • [21] M. Pollicott. Rates of mixing for potentials of summable variation. Trans. Amer. Math. Soc., to appear. CMP 98:12
  • [22] F. Schweiger. Some remarks on ergodicity and invariant measures. Michigan Math. J. 22 (1975), 123-131. MR 47:8446
  • [23] F. Schweiger. Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford University Press. MR 97h:11083
  • [24] M. Thaler. Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J. Math. 37 (1980), 303-314. MR 82f:28021
  • [25] M. Thaler. Transformations on $ [ 0,1 ]$ with infinite invariant measures. Israel J. Math. 46 (1983) 67-96. MR 85g:28020
  • [26] M. S. Waterman, Some ergodic properties of multi-dimensional $ f $-expansions. Z. Wahrsch. Verw. Gebiete 16 (1971), 77-103. MR 44:173
  • [27] M. Yuri. On a Bernoulli property for multi-dimensional mappings with finite range structure. Tokyo J. Math. 9 (1986) 457-485. MR 88d:28023
  • [28] M. Yuri. Invariant measures for certain multi-dimensional maps. Nonlinearity 7 (3) (1994), 1093-1124. MR 95c:11101
  • [29] M. Yuri. Multi-dimensional maps with infinite invariant measures and countable state sofic shifts. Indagationes Mathematicae 6 (1995), 355-383. MR 96j:58059
  • [30] M. Yuri. On the convergence to equilibrium states for certain nonhyperbolic systems. Ergodic Theory and Dynamical Systems 17, (1997) 977-1000. MR 98f:58155
  • [31] M. Yuri. Zeta functions for certain nonhyperbolic systems and topological Markov approximations. Ergodic Theory and Dynamical Systems 18 (1998), 1589-1612. CMP 99:05
  • [32] M. Yuri. Decay of correlations for certain multi-dimensional maps. Nonlinearity 9 (1996), 1439-1461. MR 97i:58100
  • [33] M. Yuri. Thermodynamic Formalism for certain nonhyperbolic maps. To appear in Ergodic Theory and Dynamical Systems.
  • [34] R. Zweimuller. Probabilistic properties of dynamical systems with infinite invariant measures, Univ. Salzburg M.Sc. thesis (1995).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11K50, 11K55, 28D05, 58F03, 58F11, 58F15

Retrieve articles in all journals with MSC (2000): 11K50, 11K55, 28D05, 58F03, 58F11, 58F15


Additional Information

Michiko Yuri
Affiliation: Department of Business Administration, Sapporo University, Nishioka, Toyohira-ku, Sapporo 062, Japan
Email: yuri@math.sci.hokudai.ac.jp, yuri@math-ext.sapporo-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-00-02579-4
Received by editor(s): February 20, 1998
Published electronically: February 14, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society