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 non-atomic, $\sigma$-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.