Julia sets that are full of holes |
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Authors: | Kimberly A Roth |
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Institution: | (1) Department of Mathematics, Juniata College, 16652 Huntingdon, PA, USA |
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Abstract: | Conclusion Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Although there
are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure
bigger than zero have been found. Shishikura’s surprising result (1998) shows that there are other Julia sets of dimension
2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of
holes, or porous, provides a bound on the upper box dimension, but this has so far been possible only for special classes
of Julia sets. Mean porosity and mean e-porosity, both found in Koskela and Rohde (1997), provide better dimension bounds;
nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used
in some cases when it is not possible to prove porosity. In the end, we do not know in general which Julia sets are porous
and which are not. In fact, forJ
R, little is known about its dimension or measure. There is much left to explore. |
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Keywords: | |
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