Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints

Details Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints

2007-04-23

arxiv.org/abs/0704.3087v2

Nonlinearity 21 (2008), 113-120

Mathematics

Dynamical Systems

10 pages

Scientific paper

10.1088/0951-7715/21/1/006

We investigate the set $I$ of parameters $\kappa$ for which the singular value of $z\mapsto e^z+\kappa$ converges to $\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in $I$ alone have dimension 2. Analogous results were known for dynamical planes of exponential maps; our result shows that this also holds in parameter space.

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