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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fractal Dimensions and Random Transformations
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by Yuri Kifer PDF
Trans. Amer. Math. Soc. 348 (1996), 2003-2038 Request permission

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
  • 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.
  • © Copyright 1996 American Mathematical Society
  • 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