Hausdorff dimension of elliptic functions with critical values approaching infinity

Details Hausdorff dimension of elliptic functions with critical values approaching infinity Hausdorff dimension of elliptic functions with critical values approaching infinity

2011-05-05

arxiv.org/abs/1105.1021v2

Mathematics

Dynamical Systems

24 pages

Scientific paper

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.

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