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A local limit theorem for a sequence of interval transformations

Published online by Cambridge University Press:  19 September 2008

P. Calderoni ,
M. Campanino  and
D. Capocaccia
P. Calderoni
Affiliation:
ZiF, Universität Bielefeld, 4800 Bielefeld, West Germany
M. Campanino
Affiliation:
ZiF, Universität Bielefeld, 4800 Bielefeld, West Germany
D. Capocaccia
Affiliation:
Instituto Matematico ‘G. Castelnuovo’, Università di Roma, 00185 Roma, Italy
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Abstract

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Let λ > 1 be a real eigenvalue of an automorphism of the two dimensional torus. We prove that for a dense, open subset of intervals the sequence where {x} denotes the fractional part of x and χ[a, b] the characteristic function of [a, b], satisfies the local limit theorem with respect to Lebesgue measure on [0, 1].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 2 , June 1985 , pp. 185 - 201
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Adler, R. L. & Weiss, B.. Similarity of automorphisms of the torus. Mem. Amer. Math. Soc. 98 (1970).Google Scholar
[2]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes in Math. 470. Springer-Verlag, 1975.CrossRefGoogle Scholar
[3]Cornfeld, I. P., Fomin, S. V. & Sinai, Ya. G.. Ergodic Theory. Springer Verlag, 1982.CrossRefGoogle Scholar
[4]Dobrushin, R. L. & Tirozzi, B.. The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54 (1977), 173–192.CrossRefGoogle Scholar
[5]Gnedenko, B. V. & Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables. Addison Wesley, 1954.Google Scholar
[6]Ibragimov, I. A.. The central limit theorem for sums of functions of independent random variables and sums of the form . Theory Probab. Appl. 12 (1967), 596–607.CrossRefGoogle Scholar
[7]Ibragimov, I. A. & Linnik, Yu. V.. Independent and Stationary Sequences of Random Variables. Noordhoff: Groningen, 1971.Google Scholar
[8]Moskvin, D. A. & Postnikov, A. G.. A local limit theorem for the distribution of fractional parts of an exponential function. Theory Probab. Appl. 23 (1978), 521–528.CrossRefGoogle Scholar
[9]nagaev, S. V.. Some limit theorems for homogeneous Markov chains. Theory Probab. Appl. 2 (1957) 378–406.CrossRefGoogle Scholar
[10]Phillipp, W.. Some metrical theorems in number theory, II. Duke Math. J. 37 (1970), 447–458.Google Scholar
[11]Rousseau-Egele, J.. Un théorème de la limite locale pour une classe de transformations monotones par morceaux. Preprint.Google Scholar
[12]Sinai, Ya. G.. Introduction to Ergodic Theory. Princeton Univ. Press, 1976.Google Scholar