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The ergodic theory and geometry of the Julia set of meromorphic functions onthe complex plane with polynomial Schwarzian derivative is investigated underthe condition that the forward trajectory of asymptotic values in the Julia setis bounded and the map $f$ restricted to its closure is expanding, the propertyrefered 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 weprove the existence of a $\sg$--finite invariant measure $\mu$ absolutelycontinuous with respect to $m$. Our main result states that $\mu$ is finite ifand 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 metricallyexact. We also establish a version of Bowen's formula showing that the exponent$h$ equals the Hausdorff dimension of the Julia set of $f$.

Ergodic properties of sub-hyperbolic functions with polynomial Schwarzian derivative