## 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 series arXiv:1105.1021, (0 pkt) Issue year 2011 Keywords in English Mathematics - Complex Variables, Mathematics - Dynamical Systems, Primary 37F35. Secondary 37F10, 30D05 URL http://arxiv.org/abs/1105.1021 Score (nominal) 0 Publication indicators WoS Citations = 0 Citation count*
 Cite