Abstract
The goal of this mainly expository paper is to develop the theory of the algebraic entropy in the basic setting of vector spaces V over a field K. Many complications encountered in more general settings do not appear at this first level. We will prove the basic properties of the algebraic entropy of linear transformations \({\phi:V \to V}\) of vector spaces and its characterization as the rank of V viewed as module over the polynomial ring K[X] through the action of \({\phi}\) . The two main theorems on the algebraic entropy, namely, the Addition Theorem and the Uniqueness Theorem, whose proofs require many efforts in more general settings, are easily deduced from the above characterization. The adjoint algebraic entropy of a linear transformation, its connection with the algebraic entropy of the adjoint map of the dual space and the dichotomy of its behavior are also illustrated.
Highlights
1 Introduction The following questions, even if formulated in a vague way, should excite curiosity in people with a basic mathematical education: Question 1.1 Given a linear transformation φ : V → V of a vector space V over a fixed field K, how chaotic is the iterated action of φ in V ? Can we measure the dynamical behavior of the linear transformation φ?
General deep results for modules over arbitrary rings have been recently obtained for the algebraic entropy
As a consequence of the Addition Theorem, we prove the counterpart of the Grassmann formula for the algebraic entropy: Corollary 5.2 Let φ : V → V be a linear transformation, and let U and W be φ-invariant subspaces of V such that V = U + W
Summary
The following questions, even if formulated in a vague way, should excite curiosity in people with a basic mathematical education: Question 1.1 Given a linear transformation φ : V → V of a vector space V over a fixed field K , how chaotic is the iterated action of φ in V ? Can we measure the dynamical behavior of the linear transformation φ?. A relevant difference between the algebraic entropy ent and its adjoint version ent is that the latter presents a dichotomy in its behavior, since it takes only values 0 and ∞ The proof of this dichotomy furnished here for vector spaces, similar to the analogous proof for Abelian groups given in [5], makes an essential use of some structure results of modules over PID’s. In the setting of Abelian groups, the adjoint algebraic entropy does not satisfy the Addition Theorem, except when one considers only bounded groups; in the present setting of vector spaces, we will prove the Addition Theorem for ent in full generality, as an easy consequence of the dichotomy of ent
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