Hausdorff dimension of elliptic functions with critical values approaching infinity
Abstract
We consider the escaping parameters in the family $\beta\wp_\Lambda$, i.e. these parameters for which the orbits of critical values of $\beta\wp_\Lambda$ approach infinity, where $\wp_\Lambda$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a nonatomic, $\sigma$finite, ergodic, conservative and invariant measure $\mu$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the $\wp_\Lambda$function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family $\beta\wp_\Lambda$, and compare it with the Hausdorff dimension of escaping set in dynamical space, proving a similarity between parameter plane and dynamical space.
 Publication:

arXiv eprints
 Pub Date:
 May 2011
 DOI:
 10.48550/arXiv.1105.1021
 arXiv:
 arXiv:1105.1021
 Bibcode:
 2011arXiv1105.1021G
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Complex Variables;
 Primary 37F35. Secondary 37F10;
 30D05
 EPrint:
 24 pages