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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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