Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 28A78, 58F15, 28A80, 60F10

Retrieve articles in all journals with MSC (1991): 28A78, 58F15, 28A80, 60F10


Additional Information

Yuri Kifer
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
Email: kifer@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01608-X
PII: S 0002-9947(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