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θ-transformations, θ-shifts and limit theorems for some Riesz-Raikov sums

Published online by Cambridge University Press:  19 September 2008

Bernard Petit
Bernard Petit
Affiliation:
Département de Mathématiques, U.F.R. des Sciences et Techniques, 6, avenue Victor Le Gorgeu, BP 809 29285 Brest Cedex, France
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Abstract

By Fourier methods, it is possible to prove central limit theorems for Riesz-Raikov sums for any real θ > 1 and for a quite large class of functions f.

The same problem is solved here for Hölder-continuous functions f and Pisot–Vijayaragavan numbers θ in a totally different way: it is shown that the question is equivalent to working with ergodic sums for suitable functions F on [0;l], and T denoting the θ-transformation x ↦ θx mod 1. In addition, limit theorems are proved on the θ-shifts, for any real θ 1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 2 , April 1996 , pp. 335 - 364
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[Be]Bertrand-Mathis, A.. Développement en base θ, répartition modulo un de la suite {xθn}, langages codes et θ-shifts. Bull. Soc. math. France. 114 (1986), 271–323.Google Scholar
[Bl]Blank, M. L.. Stochastic properties of deterministic dynamical systems. Sov. Sci. Rev. C Maths/Phys. 6 (1987), 243–271.Google Scholar
[Bo]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes Math. 470. Berlin, Springer, 1975.Google Scholar
[Br]Breiman, L.. Probability. Addison-Wesley, Reading, 1968.Google Scholar
[C]Cigler, J.. Ziffernverteilung in θ-adischen Brchen. Math. Z. 75 (1961), 8–13.Google Scholar
[CCC]Calderoni, P., Campanino, M. and Capocaccia, D.. A local limit theorem for a sequence of interval transformations. Ergod. Th. & Dynam. Sys. 5 (1985), 185–201.Google Scholar
[Fi]Fischer, R.. Ergodische Theorie von Ziffernentwicklungen und Wahrscheinlichkeitsráumen. Math. Z. 128 (1972), 217–230.Google Scholar
[Fu]Fukuyama, K.The central limit theorem for the Riesz-Raikov sums. Probab. Theory Related Fields. 100(1)(1994), 57–75.CrossRefGoogle Scholar
[GH]Guivarc'h, Y. and Hardy, J.. Théorèmes limites pour une clase de chaînes de Markov et applications aux difféomorphismes d'Anosov. Ann. last. H. P. 24(1) (1988), 73–98.Google Scholar
[Hk]Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.Google Scholar
[IM]Ionescu Tulcea, C. T. and Marinescu, G.. Théorie ergodique pour des classes d'opérateurs non completement continus. Ann. Math. 47 (1946), 140–147.Google Scholar
[IT]Ito, S. and Takahashi, Y.. Markov subshifts and realization of β-expansions. J. Math. Soc. Japan. 26(1) (1974), 33–55.Google Scholar
[Ka]Kac, M.. On the distribution of values of sums of the form Σ f(2kt). Ann. Math. 47 (1946), 33–49.Google Scholar
[Ke]Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahr. verw. Geb. 69 (1985), 461–478.Google Scholar
[Pa]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sc. Hung. 11 (1960), 401–416.Google Scholar
[Pe]Petit, B.. Le théorème limite central pour des sommes de Riesz-Raikov. Probab. Theory Related Fields. 93 (1992), 407–438.Google Scholar
[Ph]Philipp, W.. Some metrical theorems in number theory. Pacific J. Math. 20 (1983), 599–629.Google Scholar
[Ro]Rousseau-Egele, J.. Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. Ann. Prob. 11(3) (1983), 772–788.Google Scholar
[Wo]Wong, S.. Hölder continuous derivatives and ergodic theory. J. Land. Math. Soc. 22 (1980), 506–520.CrossRefGoogle Scholar