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01 February 2007 The dynamical fine structure of iterated cosine maps and a dimension paradox
Dierk Schleicher
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Duke Math. J. 136(2): 343-356 (01 February 2007). DOI: 10.1215/S0012-7094-07-13625-1


We discuss in detail the dynamics of maps zaez+be-z for which both critical orbits are strictly preperiodic. The points that converge to under iteration contain a set R consisting of uncountably many curves called rays, each connecting to a well-defined “landing point” in C, so that every point in C is either on a unique ray or the landing point of several rays.

The key features of this article are the following:

(1) this is the first example of a transcendental dynamical system, where the Julia set is all of C and the dynamics is described in detail for every point using symbolic dynamics;

(2) we get the strongest possible version (in the plane) of the “dimension paradox”: the set R of rays has Hausdorff dimension 1, and each point in C\R is connected to by one or more disjoint rays in R.

As the complement of a 1-dimensional set, C\R of course has Hausdorff dimension 2 and full Lebesgue measure


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Dierk Schleicher. "The dynamical fine structure of iterated cosine maps and a dimension paradox." Duke Math. J. 136 (2) 343 - 356, 01 February 2007.


Published: 01 February 2007
First available in Project Euclid: 21 December 2006

zbMATH: 1114.37024
MathSciNet: MR2286634
Digital Object Identifier: 10.1215/S0012-7094-07-13625-1

Primary: 37F35
Secondary: 30D05 , 37B10 , 37C45 , 37D45 , 37F10 , 37F20

Rights: Copyright © 2007 Duke University Press


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Vol.136 • No. 2 • 01 February 2007
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