The parabolic map f1/e(z)=(1/e)e |
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Authors: | Mariusz Urbański Anna Zdunik |
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Institution: | a Department of Mathematics, University ofNorth Texas, P.O. Box 311430, Denton, TX 76203-1430, USA b Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland |
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Abstract: | We consider the exponential maps ?λ : ? → ? defined by the formula ?λ (z) = λez, λ(0,1/e]. Let Jr(?λ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ?. Our main result is that the function λhλ :=HD(Jr(?λ),)), λ(0, 1/e], is continuons at the point 1/e. As a preparation for this result we deal with the map ?1/e itself. We prove that the h1/e-dimensional Hausdorff measure of Jr(?1/e) is positive and finite on each horizontal strip, and that the h1/e-dimensional packing measure of Jr(?λ) is locally infinite at each point of Jr(?λ). Our main technical devices are formed by the, associated with ?λ, maps Fλ defined on some strip P of height 2π and also associated with them tonformal measures. |
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