Hausdorff dimension of elliptic functions with critical values approaching infinity

TL;DR

This paper estimates the Hausdorff dimension of parameters for elliptic functions with critical values escaping to infinity, revealing a similarity with the dynamical space and extending understanding beyond exponential maps.

Contribution

It provides the first Hausdorff dimension estimates for escaping parameters in elliptic functions, showing a connection with the dynamical space similar to exponential maps.

Findings

Lower bounds for Hausdorff dimension of escaping parameters

Comparison between parameter space and dynamical space dimensions

Extension of results from exponential to elliptic functions

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.