Transverse properties of dynamical systems

Roughly speaking, two dynamical systems are transversally equivalent if they have the same space of orbits; a property is transverse if it is preserved under transverse equivalence. Various notions of transverse equivalence have been defined: among them, similarity of measured groupoids, Morita equivalence of locally compact groupoids, stable orbit equivalence of measure equivalence relations. After reviewing some of these notions, we pass to discussing the Morita equivalence of groupoids and give examples and applications, in particular in connection with operator algebras.