Geometry and ergodic theory of nonhyperbolic exponential maps
Authors:
Mariusz Urbanski and Anna Zdunik
Journal:
Trans. Amer. Math. Soc. 359 (2007), 39733997
MSC (2000):
Primary 37F35; Secondary 37F10, 30D05
Published electronically:
March 20, 2007
MathSciNet review:
2302520
Fulltext PDF Free Access
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Abstract: We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the dimensional Hausdorff measure of is positive and finite, and that the dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.
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 [DU2]
 M. Denker, M. Urbanski, Geometric measures for parabolic rational maps, Ergod. Th. and Dynam. Sys. 12 (1992), 5366. MR 1162398 (93d:58133)
 [EL]
 A. Eremenko, M. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Grenoble 42 (1992), 9891020 MR 1196102 (93k:30034)
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 H. Federer, Geometric Measure Theory, Springer 1969. MR 0257325 (41:1976)
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 M. Guzmán: Differentiation of integrals in . Lect. Notes in Math. 481, Springer Verlag. MR 0476978 (57:16523)
 [Ha]
 W. Hayman, The maximum modulus and valency of functions meromorphic in the unit circle, Acta Math. 86 (1951), 89257. MR 0045211 (13:546a)
 [Lyu]
 M. Lyubich, The measurable dynamics of the exponential map, Siberian Journ. Math, 28 (1987), 111127 MR 0924986 (89d:58071)
 [Ma]
 M. Martens, The existence of finite invariant measures, Applications to real onedimensional dynamics, Preprint.
 [McM]
 C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 325342 MR 0871679 (88a:30057)
 [PU]
 F. Przytycki, M. Urbanski, Fractals in the Plane  the Ergodic Theory Methods, available on the web:http://www.math.unt.edu/urbanski, to appear in Cambridge Univ. Press.
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 M. Rees, The exponential map is not recurrent, Math. Z. 191 (1986), 593598 MR 0832817 (87g:58063)
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 S.J. Taylor, C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. A.M.S. 288 (1985), 679  699. MR 0776398 (87a:28002)
 [UZ1]
 M. Urbanski, A. Zdunik, The finer geometry and dynamics of exponential family, Michigan Math. J. 51 (2003) 227  250. MR 1992945 (2004d:37068)
 [UZ2]
 M. Urbanski, A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia set of exponential family, Erg. Th. Dynam. Sys. 24 (2004), 279  315. MR 2041272 (2005d:37096)
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Additional Information
Mariusz Urbanski
Affiliation:
Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 762031430
Email:
urbanski@unt.edu
Anna Zdunik
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02097 Warszawa, Poland
Email:
A.Zdunik@mimuw.edu.pl
DOI:
http://dx.doi.org/10.1090/S0002994707041517
PII:
S 00029947(07)041517
Received by editor(s):
September 18, 2003
Received by editor(s) in revised form:
August 3, 2005
Published electronically:
March 20, 2007
Additional Notes:
The research of the first author was supported in part by the NSF Grant DMS 0400481.
The research of the second author was supported in part by the Polish KBN Grant 2 PO3A 034 25. The research of both authors was supported in part by the NSF/PAN grant INT0306004.
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American Mathematical Society
