TY - JOUR
T1 - Escaping points of exponential maps
AU - Schleicher, Dierk
AU - Zimmer, Johannes
N1 - Funding Information:
Proof. The estimates in Proposition 4.5 show immediately that for any external address s and any potential t > ts, we have z = gs(t) 2 I0(f) and z = gs(t) 2 Z(f). We also know that every escaping point is in the closure of a single ray, so I0(f) = Z(f) = I(f). It is well known that I(f) = J(f) for entire maps with nitely many singular values (this is a special case of [5, p. 344]). For our maps, we noted earlier that I(f) J(f), and since the Julia set is closed, we have I(f) J(f). Conversely, it follows immediately from Montel’s theorem that escaping points are dense in the Julia set. q Acknowledgements. This project was inspired by discussions with Bogusia Karpinska and Misha Lyubich at a Euroconference in Crete organized by Shaun Bullett, Adrien Douady and Christos Kourouniotis. We also thank Bob Devaney, Nuria Fagella, John Hubbard and Lasse Rempe for interesting discussions. We gratefully acknowledge support and encouragement by John Milnor and the Institute for Mathematical Sciences in Stony Brook. Much of this work was carried out while we held positions at the Ludwig-Maximilians-Universität München and the Technische Universität München, respectively.
PY - 2003/4
Y1 - 2003/4
N2 - The points which converge to ∞ under iteration of the maps z → λ exp(z) for λ ∈ ℂ\{0} are investigated. A complete classification of such 'escaping points' is given: they are organized in the form of differentiate curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of λ.
AB - The points which converge to ∞ under iteration of the maps z → λ exp(z) for λ ∈ ℂ\{0} are investigated. A complete classification of such 'escaping points' is given: they are organized in the form of differentiate curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ. It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of λ.
UR - http://www.scopus.com/inward/record.url?scp=0037907639&partnerID=8YFLogxK
U2 - 10.1112/S0024610702003897
DO - 10.1112/S0024610702003897
M3 - Article
AN - SCOPUS:0037907639
SN - 0024-6107
VL - 67
SP - 380
EP - 400
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
IS - 2
ER -