  In this paper we study measurable dynamics for the widest reasonable class of
smooth one dimensional maps. Three principle decompositions are described in
this class : decomposition of the global measure-theoretical attractor into
primitive ones, ergodic decomposition and Hopf decomposition. For maps with
negative Schwarzian derivative this was done in the series of papers [BL1-BL5],
but the approach to the general smooth case must be different.
