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Dynamics of linear and affine maps. (English) Zbl 1178.37029

Author’s summary: 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.

In this paper we approach the issue from a “dynamic” viewpoint, as explained in the author’s paper [J. Ramanujan Math. Soc. 22, No. 1, 35–56 (2007; Zbl 1181.22022)]. We also extend the theory to affine maps. The formulation is in terms of the action of the general 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. An important ingredient in these parametrizations is a certain flag. For a linear map T on 𝕍, let Z L (T) denote its centralizer associative 𝔽-algebra, and Z L (T) * the multiplicative group of invertible elements in Z L (T). In this situation, we associate a canonical, maximal, Z L (T)-invariant flag, and precisely describe the orbits of Z L (T) * on 𝕍. Using this approach, we strengthen the classical theory in a number of ways.

MSC:
37C99Smooth dynamical systems
15A04Linear transformations, semilinear transformations (linear algebra)
20G15Linear algebraic groups over arbitrary fields