| Abstract: |
| Many famous fractal sets (for example, Cantor set, Sierpinski gasket and so on) are defined as the limit sets of contractive iterated function systems (for short IFSs) with finitely many mappings. But, recently D. Mauldin and M. Urbanski studied limit sets
of conformal IFSs(for short CIFSs) with infinitely many mappings. And, they showed that there exists a CIFS such that the Hausdorff measure of the limit set corresponding to the Hausdorff dimension is zero and the packing measure of the limit set corresponding to the same dimension is positive. Note that the limit sets of CIFSs with finitely many mappings do not have the above properties. In this talk, we introduce an analytic family of CIFSs with infinitely many mappings related to complex continued fractions such that the limit set of each system in the family has the above strange properties and such that the Hausdorff dimension of the limit set is
a real analytic and subharmonic function of the parameter. This study is a joint work with Hiroki SUMI (Kyoto University) and Hikaru OKADA (Osaka University). |
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