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Transactions of the American Mathematical Society

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Fractal Dimensions and Random Transformations

Author: Yuri Kifer
Journal: Trans. Amer. Math. Soc. 348 (1996), 2003-2038
MSC (1991): Primary 28A78; Secondary 58F15, 28A80, 60F10
MathSciNet review: 1348865
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Abstract: I start with random base expansions of numbers from the interval $[0,1]$ and, more generally, vectors from $[0,1]^{d}$, which leads to random expanding transformations on the $d$-dimensional torus $\mathbb{T}^{d}$. As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets ``invariant'' with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley's book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.

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Additional Information

Yuri Kifer
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

Keywords: Hausdorff dimension, random transformations, repellers
Received by editor(s): November 30, 1994
Received by editor(s) in revised form: June 16, 1995
Additional Notes: Partially supported by the US-Israel Binational Science Foundation.
Article copyright: © Copyright 1996 American Mathematical Society