Hausdorff dimension of elliptic functions with critical values approaching infinity

Piotr Gałązka

Abstract

We consider the escaping parameters in the family \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$, i.e. these parameters for which the orbits of critical values of \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$ approach infinity, where \$\\textbackslash\wp\_\\textbackslash\Lambda\$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a non-atomic, \$\\textbackslash\sigma\$-finite, ergodic, conservative and invariant measure \$\\textbackslash\mu\$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the \$\\textbackslash\wp\_\\textbackslash\Lambda\$-function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family \$\\textbackslash\beta\\textbackslash\wp\_\\textbackslash\Lambda\$, and compare it with the Hausdorff dimension of escaping set in dynamical space, proving a similarity between parameter plane and dynamical space.
Author Piotr Gałązka (FMIS / DFE)
Piotr Gałązka,,
- Department of Functional Equations
Journal seriesarXiv:1105.1021, (0 pkt)
Issue year2011
Keywords in EnglishMathematics - Complex Variables, Mathematics - Dynamical Systems, Primary 37F35. Secondary 37F10, 30D05
URL http://arxiv.org/abs/1105.1021
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