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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
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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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  • [ABD] J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill, B. Weiss, Strong laws for $L$- and $U$-statistics, Trans. Amer. Math. Soc. (to appear).
  • [Bed] T. Bedford, On Weierstrass-like functions and random recurrent sets, Math. Proc. Camb. Phil. Soc. 106 (1989), 325-342. MR 91c:26010
  • [Be] A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Annalen 110 (1934), 321-330.
  • [Bi1] P. Billingsley, Hausdorff dimension in probability theory. I, II, Illinois J. Math 4 (1960), 187-209; 5 (1961), 291-298. MR 24a:1750, MR 22:11094
  • [Bi2] P. Billingsley, Ergodic Theory and Information, John Wiley, New York, 1965. MR 33:254
  • [Bi3] P. Billingsley, Hausdorff dimension: self-similarity and independent processes; cross-
    similarity and Markov processes
    , in: Statistics and Probability: A Raghu Raj Bahadur Festschrift (J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S. Prakasa Rao, eds.), Wiley Eastern Ltd, 1993, pp. 97-134.
  • [Bo] T. Bogenschutz, Entropy, pressure, and a variational principle for random dynamical systems, Random & Comp.Dyn. 1 (1992), 99-116. MR 93:28023
  • [BG1] T. Bogenschutz and V. M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random & Comp.Dyn. 1 (1992), 219-227. MR 93j:58042
  • [BG2] T. Bogenschutz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergod. Th. & Dyn. Sys. 15 (1995), 413--447. CMP 95:14
  • [BK] M. Brin and A. Katok, On local entropy, in: Geometric Dynamics, Lect. Notes in Math. 1007, Springer-Verlag, New York, 1983, pp. 30-38. MR 85c:58063
  • [Ca] H. Cajar, Billingsley Dimension in Probability Spaces, Lect. Notes in Math. 892, Springer-Verlag, Berlin, 1981. MR 84a:10055
  • [DZ] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993. MR 95a:60034
  • [Eg] H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31-36. MR 11:88
  • [Fa] K. Falconer, Fractal Geometry (Mathematical Foundations and Applications), J. Wiley & Sons, Chichester, 1990. MR 92j:28008
  • [Fu] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Th. 1 (1967), 1-49. MR 35:4369
  • [Ki1] Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), 505-524. MR 91e:60091
  • [Ki2] Y. Kifer, Equilibrium states for random expanding transformations, Random & Comp.Dyn. 1 (1992), 1-31. MR 93j:58075
  • [KK] K. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinai's Moscow Seminar on Dynamical Systems (L. A. Buninovich, B. M. Gurevich, Ya. B. Pesin, eds.), AMS Translations-Series 2, 1995.
  • [KP1] R. Kenyon and Y. Peres, Intersecting random translates of invariant Cantor sets, Invent. Math. 104 (1991), 601-629. MR 92g:28018
  • [KP2] R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. Th. & Dynam. Sys. 15 (1995).
  • [LP] L. Lovász and M. D. Plummer, Matching Theory, North-Holland, Amsterdam, 1986. MR 88b:90087
  • [LW] F. Ledrappier and P. Walters, A relativized variational principle for continuous transformations, J. London Math. Soc. 16 (1977), 568-576. MR 57:16540
  • [Mc] C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984), 1-9. MR 86h:11061
  • [MO] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. MR 81b:00002
  • [Ol] L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Mathematics, vol. 307, Longman Sci. Tech., Harlow, 1994. MR 95j:28006
  • [Pe] Y. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Stat. Phys. 71 (1993), 529-547. MR 94d:28008
  • [Pey] J. Peyrière, Calculs de dimensions de Hausdorff, Duke Math. J. 44 (1977), 591-601. MR 56:3257
  • [Ph] W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arithm. 26 (1975), 241-251. MR 52:325
  • [PW] Y. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Preprint, 1994.
  • [Ru] D. Ruelle, Bowen's formula for the Hausdorff dimension of self-similar sets, in: Scaling and Self-similarity in Physics (Progress in Physics 7), Birkhäuser, Boston, 1983, pp. 351-357. MR 85d:58051
  • [Sm] M. Smorodinsky, Singular measures and Hausdorff measures, Israel J. Math. 7 (1969), 203-206. MR 40:3589
  • [St] V. Strassen, The existence of probability measures with given marginals, Ann. of Math. Stat. 36 (1965), 423-439. MR 31:1693
  • [Wa] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153. MR 57:6371
  • [Yo] L.-S. Young, Dimension, entropy, and Lyapunov exponents, Ergod. Th. & Dyn. Sys. 2 (1982), 109-129. MR 84h:58087

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Additional Information

Yuri Kifer
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

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

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