On the existence of conditionally invariant probability measures in dynamical systems

Let T : X --> X be a measurable map defined on a Polish space X and let Y be a non-trivial subset of X. We give conditions ensuring the existence of conditionally invariant probability measures to non-absorption in Y. For dynamics which are non-singular with respect to some fixed probability measure we supply sufficient conditions for the existence of absolutely continuous conditionally invariant measures. These conditions are satisfied for a wide class of dynamical systems including systems that are Phi-mixing and Gibbs. AMS classification scheme numbers: 37A05, 28D05.