Tropical linear maps on the plane

Abstract

In this paper we fully describe all tropical linear maps in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by a real 3×3 matrix A. The map fA is continuous and piecewise-linear in the classical sense. In some particular cases, the map fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the map collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (Theorem 3). In order to study fA, we may assume that A is normal, i.e., I A 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning. On , any , some aspects of tropical linear maps have been studied in [6]. We work in , adding a geometric view and doing everything explicitly. We give precise pictures. Inspiration for this paper comes from [3,5,6,8,12,26]. We have tried to make it self-contained. Our preparatory results present noticeable relationships between the algebraic properties of a given matrix A (idempotent normal matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see Theorem 2 and Corollary 1. As a by-product, we compute all the tropical square roots of normal matrices of a certain type; see Corollary 4. This is, perhaps, a curious result in tropical algebra. Our final aim is, however, to give a precise description of the map . This is particularly easy when two tropical triangles arising from A (denoted and ) fit as much as possible. Then the action of fA is easily described on (the closure of) each cell of the cell decomposition ; see Theorem 3. Normal matrices play a crucial role in this paper. The tropical powers of normal matrices of size satisfy A n-1=A n=A n+1= . This statement can be traced back, at least, to [26], and appears later many times, such as [1,2,6,9,10]. In lemma 1, we give a direct proof of this fact, for n=3. But now the equality A 2=A 3 means that the columns of A 2 are three fixed points of fA and, in fact, any point spanned by the columns of A 2 is fixed by fA. Among 3×3 normal matrices, the idempotent ones (i.e., those satisfyingA=A 2) are particularly nice: we prove that the columns of such a matrix tropically span a set which is classically compact, connected and convex (Lemma 2 and Corollary 1). In our terminology, it is a good tropical triangle