On Iterated Function Systems with place-dependent probabilities

Bárány, Balázs

doi:10.1090/S0002-9939-2014-12193-9

On Iterated Function Systems with place-dependent probabilities
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by Balázs Bárány PDF

Proc. Amer. Math. Soc. 143 (2015), 419-432 Request permission

Corrigendum: Proc. Amer. Math. Soc. 150 (2022), 5483-5483.

Abstract:

In this paper we study a family of invariant measures of parameterized iterated function systems where the corresponding probabilities are place-dependent. We prove that the Hausdorff dimension of the measure is equal to entropy/Lyapunov exponent whenever it is less than $1$ and the measure is absolute continuous w.r.t. the Lebesgue measure if entropy/Lyapunov exponent is greater than $1$ for Lebesgue almost every parameters.

References

Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
Ai Hua Fan and Ka-Sing Lau, Iterated function system and Ruelle operator, J. Math. Anal. Appl. 231 (1999), no. 2, 319–344. MR 1669203, DOI 10.1006/jmaa.1998.6210
Tian-You Hu, Ka-Sing Lau, and Xiang-Yang Wang, On the absolute continuity of a class of invariant measures, Proc. Amer. Math. Soc. 130 (2002), no. 3, 759–767. MR 1866031, DOI 10.1090/S0002-9939-01-06363-8
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Joanna Jaroszewska, Iterated function systems with continuous place dependent probabilities, Univ. Iagel. Acta Math. 40 (2002), 137–146. MR 1962721
Joanna Jaroszewska and MichałRams, On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS, J. Stat. Phys. 132 (2008), no. 5, 907–919. MR 2430786, DOI 10.1007/s10955-008-9566-3
Anna A. Kwiecińska and Wojciech Słomczyński, Random dynamical systems arising from iterated function systems with place-dependent probabilities, Statist. Probab. Lett. 50 (2000), no. 4, 401–407. MR 1802236, DOI 10.1016/S0167-7152(00)00130-9
Ka-Sing Lau, Sze-Man Ngai, and Xiang-Yang Wang, Separation conditions for conformal iterated function systems, Monatsh. Math. 156 (2009), no. 4, 325–355. MR 2486602, DOI 10.1007/s00605-008-0052-4
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Yuval Peres and Boris Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. 3 (1996), no. 2, 231–239. MR 1386842, DOI 10.4310/MRL.1996.v3.n2.a8
Yuval Peres and Boris Solomyak, Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4065–4087. MR 1491873, DOI 10.1090/S0002-9947-98-02292-2
Tomas Persson, Absolutely continuous invariant measures for some piecewise hyperbolic affine maps, Ergodic Theory Dynam. Systems 28 (2008), no. 1, 211–228. MR 2380307, DOI 10.1017/S0143385707000442
Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergodic Theory Dynam. Systems 12 (1992), no. 1, 123–151. MR 1162404, DOI 10.1017/S0143385700006635
Mark Pollicott and Károly Simon, The Hausdorff dimension of $\lambda$-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), no. 3, 967–983. MR 1290729, DOI 10.1090/S0002-9947-1995-1290729-0
Pablo Shmerkin and Boris Solomyak, Zeros of $\{-1,0,1\}$ power series and connectedness loci for self-affine sets, Experiment. Math. 15 (2006), no. 4, 499–511. MR 2293600
K. Simon, B. Solomyak, and M. Urbański, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math. 201 (2001), no. 2, 441–478. MR 1875903, DOI 10.2140/pjm.2001.201.441
K. Simon, B. Solomyak, and M. Urbański, Invariant measures for parabolic IFS with overlaps and random continued fractions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5145–5164. MR 1852098, DOI 10.1090/S0002-9947-01-02873-2
Ivan Werner, Contractive Markov systems, J. London Math. Soc. (2) 71 (2005), no. 1, 236–258. MR 2108259, DOI 10.1112/S0024610704006088