Limit and Extended Limit Sets of Matrices in Jordan Normal Form

In this note we describe the limit and the extended limit sets of every vector for a single matrix in Jordan normal form. 1. Preliminaries and basic notions Limit and extended limit sets are in the center of interest in the study of dynamics of linear operators. To find them, even in relatively easy cases of operators, it is a difficult task. In this note we describe the limit and the extended limit sets for the simplest case which is the case of a single matrix in Jordan normal form. We use a method similar to the one used by N. H. Kuiper and J. W. Robbin in [9]. In this work Kuiper and Robbin dealt with the problem of the topological classification of linear endomorphisms and the main tool they used was the extended mixing limit sets of the exponential of the nilpotent part of a Jordan block. In the following we introduce the basic notions we use in the present work. Let X be a complex Banach space and let T : X → X be a bounded linear operator. Definition 1.1. For every x ∈ X the sets L(x) = {y ∈ X : there exists a strictly increasing sequence of positive integers {kn} such that T nx → y} J(x) = {y ∈ X : there exist a strictly increasing sequence of positive integers {kn} and a sequence {xn} ⊂ X such that xn → x and