# Geometry and ergodic theory of non-recurrent elliptic functions

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## Abstract

We explore the class of elliptic functions whose critical points all contained in the Julia set are non-recurrent and whose ω-limit sets form compact subsets of the complex plane. In particular, this class comprises hyperbolic, subhyperbolic and parabolic elliptic maps. Let*h* be the Hausdorff dimension of the Julia set of such an elliptic function*f*. We construct an atomless*h*-conformal measure*m* and show that the*h*-dimensional Hausdorff measure of the Julia set of*f* vanishes unless the Julia set is equal to the entire complex plane ℂ. The*h*-dimensional packing measure is positive and is finite if and only if there are no rationally indifferent periodic points. Furthermore, we prove the existence of a (unique up to a multiplicative constant) σ-finite*f*-invariant measure μ equivalent to*m*. The measure μ is shown to be ergodic and conservative, and we identify the set of points whose open neighborhoods all have infinite measure μ. In particular, we show that ∞ is not among them.

## Keywords

Compact Subset Meromorphic Function Ergodic Theory Elliptic Function Borel Probability Measure## Preview

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