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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 $\text{GL}\left(n\right)$, resp. the group of invertible affine maps $\text{GA}\left(n\right)$, 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 $\text{GL}\left(n\right)$ on the semigroup of linear maps. We describe a parametrization of orbits and orbit-classes of both $\text{GL}\left(n\right)$- and $\text{GA}\left(n\right)$-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 ${Z}_{L}\left(T\right)$ denote its centralizer associative $𝔽$-algebra, and ${Z}_{L}{\left(T\right)}^{*}$the multiplicative group of invertible elements in ${Z}_{L}\left(T\right)$. In this situation, we associate a canonical, maximal, ${Z}_{L}\left(T\right)$-invariant flag, and precisely describe the orbits of ${Z}_{L}{\left(T\right)}^{*}$ on $𝕍$. Using this approach, we strengthen the classical theory in a number of ways.

MSC:
 37C99 Smooth dynamical systems 15A04 Linear transformations, semilinear transformations (linear algebra) 20G15 Linear algebraic groups over arbitrary fields