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Abstract  

This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem.

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Abstract  

We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a 0, a 1, a 2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.

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Katok, A. , Invariant measures for flows on oriented surfaces, Sov. Math., Dokl. , 14 (1973), no. 3, 1104–1108. MR 48 #9771 Katok A. Invariant measures for flows on

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Kharazishvili, A. B. , Transformation Groups and Invariant Measures , World Scientific Publ. Co., London-Singapore, 1998. MR 2000g :28028 Kharazishvili A. B

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. Probab. 1 36 61 LUBOTZKY, A., Discrete groups, expanding graphs, and invariant measures , With an appendix

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