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 | References | Similar Articles | Additional Information

Abstract: I start with random base expansions of numbers from the interval and, more generally, vectors from , which leads to random expanding transformations on the -dimensional torus . 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

DOI:
https://doi.org/10.1090/S0002-9947-96-01608-X

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