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Invariant measures and asymptotics for some skew products

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Abstract

For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.

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References

  1. J. Aaronson,An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs50, American Mathematical Society, Providence, R.I, 1997.

    MATH  Google Scholar 

  2. J. Aaronson,Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Israel Journal of Mathematics33 (1979), 181–197.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Aaronson and M. Denker,Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochastic Dynamics1 (2001), no. 2 193–237.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Aaronson and M. Denker,Group extensions of Gibbs-Markov maps, to appear in Probability and Related Fields.

  5. J. Aaronson and M. Keane,The visits to zero of some deterministic random walks, Proceedings of the London Mathematical Society44 (1982), 535–553.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Aaronson and B. Weiss,On the asymptotics of a 1-parameter family of infinite measure preserving transformations, Boletim da Sociedade Brasileira de Matemática (N.S.)29 (1998), 181–193.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Babillot and F. Ledrappier,Geodesic paths and horocycle flow on abelian covers, inLie Groups and Ergodic Theory (Mumbai, 1996), Tata Institute of Fundamental Research Studies in Mathematics14, Tata Institute of Fundamental Research, Bombay, 1998, pp. 1–32.

    Google Scholar 

  8. R. Bowen and B. Marcus,Unique ergodicity for horocycle foliations, Israel Journal of Mathematics26 (1977), 43–67.

    MATH  MathSciNet  Google Scholar 

  9. R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics470, Springer-Verlag, Berlin-New York, 1975.

    MATH  Google Scholar 

  10. J.-P. Conze and M. Keane,Ergodicité d’un flot cylindrique, Séminaire de ProbabilitésI, Univeristé Rennes, Rennes, 1976.

    Google Scholar 

  11. R. Ellis,Entropy, large deviations and statistical mechanics, Grundlehren der Mathematischen Wissenschaften271, Springer-Verlag, New York-Berlin, 1985.

    MATH  Google Scholar 

  12. J. Feldman and C. C. Moore,Ergodic equivalence relations. cohomology, and von Neumann algebras. I, Transactions of the American Mathematical Society234 (1977), 289–324.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Furstenberg,Strict ergodicity and transformation of the torus, American Journal of Mathematics83 (1961), 573–601.

    Article  MATH  MathSciNet  Google Scholar 

  14. Y. Guivarc’h,Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés, Ergodic Theory and Dynamical Systems9 (1989), 433–453.

    MATH  MathSciNet  Google Scholar 

  15. Y. Guivarc’h and J. Hardy,Théorèmes limites pour une class de chaînes de Markov et applications aux difféomorphisms d’Anosov, Annales de l’Institut Henri Poincaré, Probabilites et Statistiques24 (1988), 73–98.

    MATH  MathSciNet  Google Scholar 

  16. A. Hajian, Y. Ito and S. Kakutani,Invariant measures and orbits of dissipative transformations, Advances in Mathematics9 (1972), 52–65.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Herman,Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publications Mathématiques de l’Institut des Hautes Études Scientifiques49 (1979), 5–233.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. Kaimanovich,Ergodic properties of the horocycle flow and classification of Fuchsian groups, Journal of Dynamical and Control Systems6 (2000), 21–56.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Keane,Interval exchange transformations, Mathematische Zeitschrift141 (1975), 25–31.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Ya. Khinchin,Continued Fractions, University of Chicago Press, Chicago and London, 1964.

    MATH  Google Scholar 

  21. Z. S. Kowalski,Quasi markovian transformations, Ergodic Theory and Dynamical Systems17 (1997), 885–898.

    Article  MATH  MathSciNet  Google Scholar 

  22. C. Kraaikamp and P. Liardet,Good approximations and continued fractions, Proceedings of the American Mathematical Society112 (1991), 303–309.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974.

    MATH  Google Scholar 

  24. D. Maharam,Incompressible transformations, Forum Mathematicum56 (1964), 35–50.

    MATH  MathSciNet  Google Scholar 

  25. W. de Melo and S. van Strien,One-dimensional Dynamics, Ergebnisse Math. u. Grenzgeb. (3)25, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  26. H. Nakada,Piecewise linear homeomorphisms of type IIIand the ergodicity of cylinder flows, Kodai Mathematical Seminar Reports7 (1982), 29–40.

    MATH  MathSciNet  Google Scholar 

  27. H. Nakada,On a classification of PL-homeomorphisms of a circle, inProbability Theory and Mathematical Statistics (Tblisi 1982), Lecture Notes in Mathematics,1021, Springer, Berlin, Heidelberg, New York, 1983, pp. 474–480.

    Chapter  Google Scholar 

  28. W. Parry,Compact abelian group extensions of discrete dynamical systems, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete13 (1969), 95–113.

    Article  MATH  MathSciNet  Google Scholar 

  29. K. Petersen and K. Schmidt,Symmetric Gibbs measures, Transactions of the American Mathematical Society349 (1997), 2775–2811.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Pollicott,ℤ d-covers of horosphere foliations, Discrete Continuous Dynamical Systems6 (2000), 147–154.

    Article  MATH  MathSciNet  Google Scholar 

  31. D. Ruelle,Thermodynamic Formalism (The Mathematical Structures of Classical Equilibrium Statistical Mechanics), inEncyclopedia of Mathematics and its Applications 5, Addison-Wesley, Reading, Mass., 1978.

    Google Scholar 

  32. D. Ruelle,Statistical mechanics of a one-dimensional lattice gas, Communications in Mathematical Physics9 (1968), 267–278.

    Article  MATH  MathSciNet  Google Scholar 

  33. K. Schmidt,Cocycles of Ergodic Transformation Groups, Lecture Notes in Mathematics1, MacMillan Co. of India, 1977.

  34. K. Schmidt,Unique ergodicity and related problems, inErgodic Theory, Proceedings Oberwolfach 1978, Lecture Notes in Mathematics729, Springer-Verlag, Berlin-New York, 1979, pp. 188–198.

    Google Scholar 

  35. M. Stewart,Irregularities of uniform distribution, Acta Mathematica Academiae Scientiarum Hungaricae37 (1981), 185–221.

    Article  MATH  MathSciNet  Google Scholar 

  36. W. A. Veech,Topological dynamics, Bulletin of the American Mathematical Society83 (1977), 775–830.

    MATH  MathSciNet  Google Scholar 

  37. A. M. Vershik,A theorem on Markov periodic approximation in ergodic theory (Russian), inBoundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 14, Zapiski Nauchnykh Seminarov Leningrad. Otdel., Matematicheskii Institut Steklov (LOMI)115 (1982), 72–82.

  38. A. M. Vershik,A theorem on the periodic Markov approximation in ergodic theory, Journal of Soviet Mathematics28 (1985), 667–674.

    Article  MATH  Google Scholar 

  39. A. M. Vershik,A new model of the ergodic transformations, inDynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publications,23, PWN, Warsaw, 1989, pp. 381–384.

    Google Scholar 

  40. R. L. Wheeden and A. Zygmund,Measure and Integral. An Introduction to Real Analysis, Pure and Applied Mathematics,43, Marcel Dekker, Inc., New York-Basel, 1977.

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University Ramat Aviv, 69978, Tel Aviv, Israel

    Jon Aaronson

  2. Department of Mathematics, Keio University, Hiyoshi 3-14-1 Kohoku, 223, Yokohama, Japan

    Hitoshi Nakada

  3. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    Omri Sarig

  4. Department of Mathematics, University of Washington, Box 354350, 98195-4350, Seattle, WA, USA

    Rita Solomyak

Authors
  1. Jon Aaronson
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  2. Hitoshi Nakada
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  3. Omri Sarig
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  4. Rita Solomyak
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Aaronson, J., Nakada, H., Sarig, O. et al. Invariant measures and asymptotics for some skew products. Isr. J. Math. 128, 93–134 (2002). https://doi.org/10.1007/BF02785420

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