Invariant measures for parabolic IFS with overlaps and random continued fractions

Simon, K.; Solomyak, B.; Urbański, M.

doi:10.1090/S0002-9947-01-02873-2

Invariant measures for parabolic IFS with overlaps and random continued fractions
HTML articles powered by AMS MathViewer

by K. Simon, B. Solomyak and M. Urbański PDF

Trans. Amer. Math. Soc. 353 (2001), 5145-5164 Request permission

Abstract:

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no “overlaps.” We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

References

Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
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
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
B. Hunt, Dimensions of attractors of nonlinear iterated function systems, Preprint.
Robert Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155. MR 248779, DOI 10.1112/S0025579300002503
John R. Kinney and Tom S. Pitcher, The dimension of some sets defined in terms of $f$-expansions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965/66), 293–315. MR 198515, DOI 10.1007/BF00539116
F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 305–396 (French). MR 876081, DOI 10.1007/BFb0099434
R. Lyons, Singularity of some random continued fractions, J. Theor. Prob. 13 (2000), 535–545.
Ricardo Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86–117. MR 961095, DOI 10.1007/BFb0083068
Anthony Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 451–459 (1982). MR 662737, DOI 10.1017/s0143385700001371
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
D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergodic Th. and Dynam. Sys. 20 (2000), 1423–1447.
Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193–251. MR 1749437, DOI 10.1215/S0012-7094-00-10222-0
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
Steve Pincus, Singular stationary measures are not always fractal, J. Theoret. Probab. 7 (1994), no. 1, 199–208. MR 1256399, DOI 10.1007/BF02213368
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
Timothy D. Sauer and James A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 941–956. MR 1468109, DOI 10.1017/S0143385797086252
K. Simon and B. Solomyak, Hausdorff dimension for horseshoes in $\mathbb {R}^3$, Ergodic Th. and Dynam. Sys. 19 (1999), 1343–1363.
K. Simon, B. Solomyak, and M. Urbański, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math., to appear.
K. Simon, B. Solomyak, and M. Urbański, Parabolic iterated function systems with overlaps II: invariant measures, Preprint.
Boris Solomyak, On the random series $\sum \pm \lambda ^n$ (an Erdős problem), Ann. of Math. (2) 142 (1995), no. 3, 611–625. MR 1356783, DOI 10.2307/2118556
Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531–546. MR 1636589, DOI 10.1017/S0305004198002680
B. Solomyak and M. Urbański, $L^q$ densities for measures associated with parabolic IFS with overlaps, Indiana Univ. Math. J., to appear.
Mariusz Urbański, Parabolic Cantor sets, Fund. Math. 151 (1996), no. 3, 241–277. MR 1424576
Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI 10.1017/s0143385700009615