Geometry and ergodic theory of non-hyperbolic exponential maps

Authors:
Mariusz Urbanski and Anna Zdunik

Journal:
Trans. Amer. Math. Soc. **359** (2007), 3973-3997

MSC (2000):
Primary 37F35; Secondary 37F10, 30D05

DOI:
https://doi.org/10.1090/S0002-9947-07-04151-7

Published electronically:
March 20, 2007

MathSciNet review:
2302520

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Abstract | References | Similar Articles | Additional Information

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.

**[Bi]**P. Billingsley, Convergence of Probability Measures, Wiley, 2nd Edition, 1999. MR**1700749 (2000e:60008)****[DU1]**M. Denker, M. Urbanski, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365-384. MR**1107011 (92f:58097)****[DU2]**M. Denker, M. Urbanski, Geometric measures for parabolic rational maps, Ergod. Th. and Dynam. Sys. 12 (1992), 53-66. MR**1162398 (93d:58133)****[EL]**A. Eremenko, M. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Grenoble 42 (1992), 989-1020 MR**1196102 (93k:30034)****[Fe]**H. Federer, Geometric Measure Theory, Springer 1969. MR**0257325 (41:1976)****[Gu]**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), 89-257. MR**0045211 (13:546a)****[Lyu]**M. Lyubich, The measurable dynamics of the exponential map, Siberian Journ. Math, 28 (1987), 111-127 MR**0924986 (89d:58071)****[Ma]**M. Martens, The existence of -finite invariant measures, Applications to real one-dimensional dynamics, Preprint.**[McM]**C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 325-342 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.**[Re]**M. Rees, The exponential map is not recurrent, Math. Z. 191 (1986), 593-598 MR**0832817 (87g:58063)****[TT]**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)****[We]**Qiu Weiyuan, Hausdorff dimension of the M-set of , Acta Math. Sinica 10 (1994), 362 - 386 MR**1416147 (97g:30024)**

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Additional Information

**Mariusz Urbanski**

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

Email:
urbanski@unt.edu

**Anna Zdunik**

Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland

Email:
A.Zdunik@mimuw.edu.pl

DOI:
https://doi.org/10.1090/S0002-9947-07-04151-7

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 INT-0306004.

Article copyright:
© Copyright 2007
American Mathematical Society