Abstract
Suppose {f 1,...,f m } is a set of Lipschitz maps of ℝd. We form the iterated function system (IFS) by independently choosing the maps so that the map f i is chosen with probability p i (∑m i=1 p i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ℝd and as a corollary show that the measure will be singular if the modulus of the entropy ∑ i p i log p i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of ℝ.
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REFERENCES
Alexander, J. C., and Zagier, D. (1991). The entropy of a certain infinitely convolved Bernoulli measure. J.London Math.Soc. 44, 121-134.
Barnsley, M. F., and Elton, J. H. (1988). A new class of Markov processes for image encoding. Adv.Appl.Prob. 20, 14-32.
Campbell, K. M. (1997). Observational noise in skew product systems. Phys.D 107, 43-56.
Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982). Ergodic Theory, New York/ Heidelberg/Berlin, Springer-Verlag.
Derriennic, Y. (1980). Quelques applications du théore`me ergodique sous-additif. Astérisque 74, 183-201.
Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45-76.
Dubins, L. E., and Freedman, D. A. (1966). Invariant probabilities for certain Markov processes. Ann.Math.Stat. 32, 837-848.
Erdös, P. (1939). On a family of symmetric Bernoulli convolutions. Amer.J.Math. 61, 974-975.
Falconer, K. (1997). Techniques in Fractal Geometry, Wiley.
Garsia, A. (1962). Arithmetic properties of Bernoulli convolutions. Trans.Amer.Math.Soc. 102, 409-432.
Kaimanovich, V. (2000). The Poisson formula for groups with hyperbolic properties. Ann.of Math. 152 (2), 659-692.
Kaimanovich, V. (1998). Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam.Systems 18, 631-660.
Kaimanovich, V., and Vershik, A. (1983). Random walks on discrete group: Boundary and entropy, Ann.Prob. 11, 457-490.
Lalley, S. (1998). Random series in powers of algebraic numbers: Hausdorff dimension of the limit distribution, J.London Math.Soc. 57, 629-654.
Lau, K.-S., Ngai, S.-M., and Rao, H. (1999). Iterated function systems with overlaps and self-similar measures, J.London Math.Soc. (to appear).
Lyons, R. (2000). Singularity of some random continued fractions, J.Theor.Prob. 13, 535-545.
Peres, Y., Schlag, W., and Solomyak, B. (2000). Sixty years of Bernoulli convolutions, Fractals and Stochastics II. In Bandt, C., Graf, S., and Zaehle, M. (eds.), Progress in Probability, Vol. 46, Birkhauser, pp. 39-65.
Peres, Y., and Solomyak, B. (1998). Self-similar measures and intersections of Cantor sets, Trans.Amer.Math.Soc. 350, 4065-4087.
Pincus, S. (1994). Singular stationary measures are not always fractal, J.Theor.Prob. 7, 199-208.
Przytycki, F., and Urbański, M. (1989) On the Hausdorff dimension of some fractal sets, Studia Math. 93, 155-186.
Sidorov, N., and Vershik, A. (1998). Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems, Monatsh.Math. 126, 215-261.
Simon, K., Solomyak, B., and Urbański, M. (2001). Invariant measures for parabolic IFS with overlaps and random continued fractions, Trans.Amer.Math.Soc. 353, 5145-5164.
Solomyak, B. (1995). On the random series Σ±λi(an Erdös problem), Ann.of Math. 142, 611-625.
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Nicol, M., Sidorov, N. & Broomhead, D. On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average. Journal of Theoretical Probability 15, 715–730 (2002). https://doi.org/10.1023/A:1016224000145
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DOI: https://doi.org/10.1023/A:1016224000145