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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: March 20, 2007
MathSciNet review: 2302520
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Abstract: We deal with all the maps from the exponential family $ \{\lambda e^z\}$ such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters $ \lambda\in (1/e,+\infty)$ are included. We introduce as our main technical devices the projection $ F_{\lambda}$ of the map $ f_{\lambda}$ to the infinite cylinder $ Q=\mathbb{C}/2\pi i\mathbb{Z}$ and an appropriate conformal measure $ m$. We prove that $ J_r(F_\lambda)$, essentially the set of points in $ Q$ returning infinitely often to a compact region of $ Q$ disjoint from the orbit of $ 0\in Q$, has the Hausdorff dimension $ h_\lambda\in (1,2)$, that the $ h_\lambda$-dimensional Hausdorff measure of $ J_r(F_\lambda)$ is positive and finite, and that the $ h_\lambda$-dimensional packing measure is locally infinite at each point of $ J_r(F_\lambda)$. We also prove the existence and uniqueness of a Borel probability $ F_\lambda$-invariant ergodic measure equivalent to the conformal measure $ m$. As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the $ \omega$-limit set (under $ f_\lambda$) of Lebesgue almost every point in $ \mathbb{C}$, coincides with the orbit of zero under the map $ f_\lambda$. Finally we show that the the function $ \lambda\mapsto h_\lambda$, $ \lambda\in (1/e,+\infty)$, is continuous.


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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: http://dx.doi.org/10.1090/S0002-9947-07-04151-7
PII: S 0002-9947(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