Invariant measures for parabolic IFS with overlaps and random continued fractions

Authors:
K. Simon, B. Solomyak and M. Urbanski

Journal:
Trans. Amer. Math. Soc. **353** (2001), 5145-5164

MSC (2000):
Primary 37L30; Secondary 60G30

Published electronically:
July 12, 2001

MathSciNet review:
1852098

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[BL]**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****[Eg]**H. G. Eggleston,*The fractional dimension of a set defined by decimal properties*, Quart. J. Math., Oxford Ser.**20**(1949), 31–36. MR**0031026****[E]**Paul Erdös,*On a family of symmetric Bernoulli convolutions*, Amer. J. Math.**61**(1939), 974–976. MR**0000311****[F1]**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****[F2]**Kenneth Falconer,*Techniques in fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1997. MR**1449135****[H]**B. Hunt, Dimensions of attractors of nonlinear iterated function systems, Preprint.**[Ka]**Robert Kaufman,*On Hausdorff dimension of projections*, Mathematika**15**(1968), 153–155. MR**0248779****[KP]**John R. Kinney and Tom S. Pitcher,*The dimension of some sets defined in terms of 𝑓-expansions*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**4**(1965/1966), 293–315. MR**0198515****[L]**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**, 10.1007/BFb0099434**[Ly]**R. Lyons, Singularity of some random continued fractions,*J. Theor. Prob.***13**(2000), 535-545. CMP**2000:17****[Ma]**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**, 10.1007/BFb0083068**[Mn]**Anthony Manning,*A relation between Lyapunov exponents, Hausdorff dimension and entropy*, Ergodic Theory Dynamical Systems**1**(1981), no. 4, 451–459 (1982). MR**662737****[Mat]**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****[MU]**D. Mauldin and M. Urbanski, Parabolic iterated function systems,*Ergodic Th. and Dynam. Sys.***20**(2000), 1423-1447. CMP**2001:02****[PSc]**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**, 10.1215/S0012-7094-00-10222-0**[PSo1]**Yuval Peres and Boris Solomyak,*Absolute continuity of Bernoulli convolutions, a simple proof*, Math. Res. Lett.**3**(1996), no. 2, 231–239. MR**1386842**, 10.4310/MRL.1996.v3.n2.a8**[PSo2]**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**, 10.1090/S0002-9947-98-02292-2**[Pi]**Steve Pincus,*Singular stationary measures are not always fractal*, J. Theoret. Probab.**7**(1994), no. 1, 199–208. MR**1256399**, 10.1007/BF02213368**[PoS]**Mark Pollicott and Károly Simon,*The Hausdorff dimension of 𝜆-expansions with deleted digits*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 967–983. MR**1290729**, 10.1090/S0002-9947-1995-1290729-0**[SY]**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**, 10.1017/S0143385797086252**[SSo]**K. Simon and B. Solomyak, Hausdorff dimension for horseshoes in ,*Ergodic Th. and Dynam. Sys.***19**(1999), 1343-1363. CMP**2000:04****[SSU1]**K. Simon, B. Solomyak, and M. Urbanski, Hausdorff dimension of limit sets for parabolic IFS with overlaps,*Pacific J. Math.*, to appear.**[SSU2]**K. Simon, B. Solomyak, and M. Urbanski, Parabolic iterated function systems with overlaps II: invariant measures, Preprint.**[So1]**Boris Solomyak,*On the random series ∑±𝜆ⁿ (an Erdős problem)*, Ann. of Math. (2)**142**(1995), no. 3, 611–625. MR**1356783**, 10.2307/2118556**[So2]**Boris Solomyak,*Measure and dimension for some fractal families*, Math. Proc. Cambridge Philos. Soc.**124**(1998), no. 3, 531–546. MR**1636589**, 10.1017/S0305004198002680**[SU]**B. Solomyak and M. Urbanski, densities for measures associated with parabolic IFS with overlaps,*Indiana Univ. Math. J.*, to appear.**[U]**Mariusz Urbański,*Parabolic Cantor sets*, Fund. Math.**151**(1996), no. 3, 241–277. MR**1424576****[Y]**Lai Sang Young,*Dimension, entropy and Lyapunov exponents*, Ergodic Theory Dynamical Systems**2**(1982), no. 1, 109–124. MR**684248**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37L30,
60G30

Retrieve articles in all journals with MSC (2000): 37L30, 60G30

Additional Information

**K. Simon**

Affiliation:
Department of Stochastics, Institute of Mathematics, Technical University of Budapest, P.O. Box 91, 1521 Budapest, Hungary

Email:
simonk@math.bme.hu

**B. Solomyak**

Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350

Email:
solomyak@math.washington.edu

**M. Urbanski**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203-1430

Email:
urbanski@unt.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02873-2

Keywords:
Iterated function systems,
parabolic maps,
random continued fractions

Received by editor(s):
January 17, 2000

Received by editor(s) in revised form:
December 18, 2000

Published electronically:
July 12, 2001

Additional Notes:
Research of Simon was supported in part by the OTKA foundation grant F019099. Research of Solomyak was supported in part by the Fulbright foundation and the NSF grant DMS 9800786. Research of Urbański was supported in part by the NSF grant DMS 9801583

Article copyright:
© Copyright 2001
American Mathematical Society