The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.
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