Ergodic properties of sub-hyperbolic functions with polynomial Schwarzian derivative

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. 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 $\sg$--finite invariant measure $\mu$ absolutely continuous with respect to $m$. Our main result states that $\mu$ is finite if and only if the order $\rho$ of the function $f$ satisfies the condition $h>3\frac{\rho}{\rho +1}$. 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 $f$.