## Fractal Dimensions and Random Transformations

HTML articles powered by AMS MathViewer

- by Yuri Kifer
- Trans. Amer. Math. Soc.
**348**(1996), 2003-2038 - DOI: https://doi.org/10.1090/S0002-9947-96-01608-X
- PDF | 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.## References

- 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). - Tim Bedford,
*On Weierstrass-like functions and random recurrent sets*, Math. Proc. Cambridge Philos. Soc.**106**(1989), no. 2, 325–342. MR**1002545**, DOI 10.1017/S0305004100078142 - A. S. Besicovitch,
*On the sum of digits of real numbers represented in the dyadic system*, Math. Annalen**110**(1934), 321-330. - Patrick Billingsley,
*Hausdorff dimension in probability theory*, Illinois J. Math.**4**(1960), 187–209. MR**131903** - Patrick Billingsley,
*Ergodic theory and information*, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0192027** - 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. - Erich Rothe,
*Topological proofs of uniqueness theorems in the theory of differential and integral equations*, Bull. Amer. Math. Soc.**45**(1939), 606–613. MR**93**, DOI 10.1090/S0002-9904-1939-07048-1 - Thomas Bogenschütz and Volker Matthias Gundlach,
*Symbolic dynamics for expanding random dynamical systems*, Random Comput. Dynam.**1**(1992/93), no. 2, 219–227. MR**1186374** - T. Bogenschutz and V. M. Gundlach,
*Ruelle’s transfer operator for random subshifts of finite type*, Ergod. Th. & Dyn. Sys.**15**(1995), 413–447. - M. Brin and A. Katok,
*On local entropy*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 30–38. MR**730261**, DOI 10.1007/BFb0061408 - Helmut Cajar,
*Billingsley dimension in probability spaces*, Lecture Notes in Mathematics, vol. 892, Springer-Verlag, Berlin-New York, 1981. MR**654147** - Amir Dembo and Ofer Zeitouni,
*Large deviations techniques and applications*, Jones and Bartlett Publishers, Boston, MA, 1993. MR**1202429** - Morgan Ward and R. P. Dilworth,
*The lattice theory of ova*, Ann. of Math. (2)**40**(1939), 600–608. MR**11**, DOI 10.2307/1968944 - Kenneth Falconer,
*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677** - Harry Furstenberg,
*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**213508**, DOI 10.1007/BF01692494 - Yuri Kifer,
*Large deviations in dynamical systems and stochastic processes*, Trans. Amer. Math. Soc.**321**(1990), no. 2, 505–524. MR**1025756**, DOI 10.1090/S0002-9947-1990-1025756-7 - Yuri Kifer,
*Equilibrium states for random expanding transformations*, Random Comput. Dynam.**1**(1992/93), no. 1, 1–31. MR**1181378** - 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. - Richard Kenyon and Yuval Peres,
*Intersecting random translates of invariant Cantor sets*, Invent. Math.**104**(1991), no. 3, 601–629. MR**1106751**, DOI 10.1007/BF01245092 - R. Kenyon and Y. Peres,
*Measures of full dimension on affine-invariant sets*, Ergod. Th. & Dynam. Sys.**15**(1995). - L. Lovász and M. D. Plummer,
*Matching theory*, North-Holland Mathematics Studies, vol. 121, North-Holland Publishing Co., Amsterdam; North-Holland Publishing Co., Amsterdam, 1986. Annals of Discrete Mathematics, 29. MR**859549** - François Ledrappier and Peter Walters,
*A relativised variational principle for continuous transformations*, J. London Math. Soc. (2)**16**(1977), no. 3, 568–576. MR**476995**, DOI 10.1112/jlms/s2-16.3.568 - Curt McMullen,
*The Hausdorff dimension of general Sierpiński carpets*, Nagoya Math. J.**96**(1984), 1–9. MR**771063**, DOI 10.1017/S0027763000021085 - Albert W. Marshall and Ingram Olkin,
*Inequalities: theory of majorization and its applications*, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**552278** - Lars Olsen,
*Random geometrically graph directed self-similar multifractals*, Pitman Research Notes in Mathematics Series, vol. 307, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. MR**1297123**, DOI 10.2307/4351476 - Ya. B. Pesin,
*On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions*, J. Statist. Phys.**71**(1993), no. 3-4, 529–547. MR**1219021**, DOI 10.1007/BF01058436 - Jacques Peyrière,
*Calculs de dimensions de Hausdorff*, Duke Math. J.**44**(1977), no. 3, 591–601. MR**444911** - Walter Philipp,
*Limit theorems for lacunary series and uniform distribution $\textrm {mod}\ 1$*, Acta Arith.**26**(1974/75), no. 3, 241–251. MR**379420**, DOI 10.4064/aa-26-3-241-251 - Y. Pesin and H. Weiss,
*On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture*, Preprint, 1994. - David Ruelle,
*Bowen’s formula for the Hausdorff dimension of self-similar sets*, Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982) Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, pp. 351–358. MR**733478** - M. Smorodinsky,
*Singular measures and Hausdorff measures*, Israel J. Math.**7**(1969), 203–206. MR**250350**, DOI 10.1007/BF02787612 - V. Strassen,
*The existence of probability measures with given marginals*, Ann. Math. Statist.**36**(1965), 423–439. MR**177430**, DOI 10.1214/aoms/1177700153 - Peter Walters,
*Invariant measures and equilibrium states for some mappings which expand distances*, Trans. Amer. Math. Soc.**236**(1978), 121–153. MR**466493**, DOI 10.1090/S0002-9947-1978-0466493-1 - Lai Sang Young,
*Dimension, entropy and Lyapunov exponents*, Ergodic Theory Dynam. Systems**2**(1982), no. 1, 109–124. MR**684248**, DOI 10.1017/s0143385700009615

## Bibliographic 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