Dynamics of Linear and Affine Maps

The well-known theory of the “rational canonical form of an operator” describes the invariant factors, or equivalently, elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space V over a given field F. A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The viewpoint is that of finitely generated modules over a PID, cf. for example [J], ch. 3. In this paper we approach the issue from a “dynamic” viewpoint. We also extend the theory to affine maps. The formulation is in terms of the action of the geneal linear group GL(n), resp. the group of invertible affine maps GA(n), on the semigroup of all linear, resp. affine, maps by conjugacy. The theory of rational canonical forms is connected with the orbits, and the Frobenius’ theory with the orbit-classes, of the action of GL(n) on the semigroup of linear maps. We describe a parametrization of orbits and orbit-classes of both GL(n)and GA(n)-actions, and also provide a parametrization of all affine maps themselves, which is independent of the choices of linear or affine co-ordinate systems, cf. sections 7, 8, 9. An important ingredient in these parametrizations is a certain flag. For a linear map T on V, let ZL(T ) denote its centralizer associative F-algebra, and ZL(T ) ∗ the multiplicative group of invertible elements in ZL(T ). In this situation, we associate a canonical, maximal, ZL(T )-invariant flag, and precisely describe the orbits of ZL(T ) ∗ on V, cf. section 3. Using this approach, we strengthen the classical theory in a number of ways.