  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 $f$.
