On the mapping of jet spaces

Any locally invertible morphism of a finite-order jet space is either a prolonged point transformation or a prolonged Lie's contact transformation (the LieBäcklund theorem). We recall this classical result with a simple proof and moreover determine explicit formulae even for all (not necessarily invertible) morphisms of finite-order jet spaces. Examples of generalized (higher-order) contact transformations of jets that destroy all finite-order jet subspaces are stated with comments. © 2010 The Author(s).