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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-96-01608-X
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
Email: kifer@math.huji.ac.il

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