Finer geometric rigidity of limit sets of conformal IFS

Authors:
Volker Mayer and Mariusz Urbanski

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3695-3702

MSC (2000):
Primary 37D45, 37D20, 28Exx

DOI:
https://doi.org/10.1090/S0002-9939-03-07216-2

Published electronically:
July 17, 2003

MathSciNet review:
1998176

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Abstract: We consider infinite conformal iterated function systems in the phase space with . Let be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of exceeds the topological dimension of the closure of or else the closure of is a proper compact subset of either a geometric sphere or an affine subspace of dimension . A similar dichotomy holds for conformal expanding repellers.

**[BP]**R. Benedetti and C. Petronio,*Lectures on Hyperbolic Geometry*, Springer-Verlag, Berlin, 1992. MR**94e:57015****[Bo]**R. Bowen,*Hausdorff dimension of quasi-circles*, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11-25. MR**81g:57023****[Fe]**H. Federer,*Dimension and measure*, Trans. Amer. Math. Soc. 62 (1947), 536-547. MR**9:339g****[HW]**W. Hurewicz and H. Wallman,*Dimension Theory*, Princeton University Press, 1941. MR**3:312b****[Ma1]**P. Mattila,*On the structure of self-similar fractals*, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 189-192. MR**84j:28011****[Ma2]**P. Mattila,*Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability*, Cambridge University Press, 1995. MR**96h:28006****[MMU]**D. Mauldin, V. Mayer, and M. Urbanski,*Rigidity of connected limit sets of conformal IFSs*, Michigan Math. J. 49 (2001), 451-458. MR**2002j:37057****[MU1]**R. D. Mauldin and M. Urbanski,*Dimensions and measures in infinite iterated function systems*, Proc. London Math. Soc. (3) 73 (1996), 105-154. MR**97c:28020****[MU2]**R. D. Mauldin and M. Urbanski,*Conformal repellors with dimension one are Jordan curves*, Pacific Journal of Math. 166 (1994), 85-97. MR**95k:58099****[Pr]**F. Przytycki,*On holomorphic perturbations of*, Bull. Polish Acad. Sci. Math. 34 (1986), 127-132. MR**88a:58108****[PUZ]**F. Przytycki, M. Urbanski, and A. Zdunik,*Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps,*I, Ann. Math. 130 (1989), 1-40. MR**91i:58115****[Ru]**D. Ruelle,*Repellers for real analytic maps,*Ergodic Theory and Dynam. Syst. 2 (1982), 99-107. MR**84f:58095****[Su]**D. Sullivan,*Seminar on conformal and hyperbolic geometry by D. P. Sullivan*(Notes by M. Baker and J. Seade), preprint IHES (1982).**[U1]**M. Urbanski,*On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point*, Studia Math. 97 (1991), 167-188. MR**93a:58146****[U2]**M. Urbanski,*Hausdorff measures versus equilibrium states of conformal infinite iterated function systems*, Periodica Math. Hungar., 37 (1998), 153-205. MR**2001k:28012****[U3]**M. Urbanski,*Rigidity of multi-dimensional conformal iterated function systems*, Nonlinearity 14 (2001), 1593-1610. MR**2003c:37031****[UV]**M. Urbanski and A. Volberg,*A rigidity theorem in complex dynamics*, in Fractal Geometry and Stochastics, Progress in Probability 37, Birkhäuser-Verlag (1995). MR**98b:58144****[Va]**J. Väisälä,*Lectures on n-dimensional quasiconformal mappings,*Lecture Notes in Math. 229, Springer-Verlag, 1971. MR**56:12260****[Z1]**A. Zdunik,*Parabolic orbifolds and the dimension of the maximal measure for rational maps*, Invent. Math. 99 (1990), 627-649. MR**90m:58120****[Z2]**A. Zdunik,*Harmonic measure versus Hausdorff measures on repellors for holomorphic maps*, Trans. Amer. Math. Soc. 326 (1991), 633-652. MR**91k:58071**

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Additional Information

**Volker Mayer**

Affiliation:
Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France

Email:
volker.mayer@univ-lille1.fr

**Mariusz Urbanski**

Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430

Email:
urbanski@unt.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07216-2

Received by editor(s):
November 18, 2001

Published electronically:
July 17, 2003

Additional Notes:
The second author was supported in part by the NSF Grant no. DMS 0100078

Communicated by:
Michael Handel

Article copyright:
© Copyright 2003
American Mathematical Society