Dynamics of Linear and Affine Maps

TL;DR

This paper extends the classical theory of linear operators to affine maps, providing a dynamic perspective and a parametrization of orbits under conjugation by the general linear and affine groups.

Contribution

It introduces a dynamic approach to the theory of linear and affine maps and offers a new parametrization of their conjugacy classes and orbits.

Findings

Parametrization of orbits under conjugation by GL and affine groups

Extension of invariant factor theory to affine maps

Description of all affine maps via new parametrization

Abstract

The well-known theory of "rational canonical form of an operator" describes the invariant factors, or 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, as explained in [K]. We also extend the theory to affine maps. The formulation is in terms of the action of the geneal linear group, or the group of invertible affine maps, on the semigroup of all linear resp. affine maps by conjugacy. We describe a parametrization of orbits and orbit-classes under this action, and also provide a parametrization of all affine maps themselves.