We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps...

Bowen’s formula relates the Hausdorff dimension of a conformal repeller to the zero of a ‘pressure’ function. We present an elementary, self-contained proof which bypasses measure theory and the Thermodynamic Formalism to show that Bowen’s formula holds for C conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding ...

Ergodic properties of rational maps are studied, generalising the work of F. Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold...