2007/09/10 by Ravi S. Kulkarni, Kulkarni, Ravi S.
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Numerical methods for differential equations #Quantum chaos and dynamical systems #math.DS #math.GR
paper · pdf · doi:10.48550/arxiv.0709.1353
arxiv created 2007/09/10 · openalex publication_date 2007/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.