Thurston's pseudoAnosov (pA) maps play a central role in low-dimensional
topology and dynamics. After reviewing their definition and
basic properties we discuss a generalization motivated by a family
of sphere homeomorphisms derived from the tent family. The generalization
retains most of the dynamical properties and replaces the pA invariant
measured foliations with invariant measured turbulations. These consist of
a full measure subset of disjoint immersed lines each carrying a nice measure.