Adjoint algebraic entropy

Abstract

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove the Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact Abelian group, having finite positive algebraic entropy, is discontinuous

A soft introduction to algebraic entropy

“…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 .…”

“…3. We omit the proofs, which are a simplified version of the proofs given in [5] and which can be verified by the reader as a straightforward application of the definition, imitating the proofs of the corresponding properties of the algebraic entropy. The following property shows that the algebraic entropy is monotone under restrictions to subspaces and quotients.…”

“…Applying Theorem 6.15, it is now possible to give a short direct proof of the Addition Theorem for the adjoint algebraic entropy. If one would avoid the use of Theorem 6.15, one can prove it by applying the above properties of the duality, the Addition Theorem for the algebraic entropy and Theorem 6.12; this proof was given for example in [5]. …”

“…6, we investigate the adjoint algebraic entropy of a linear transformation φ : V → V , that is, ent (φ), which was introduced for endomorphisms of Abelian groups in [5] and studied also in [11]. After providing its basic properties, we will prove the main formula ent (φ) = ent(φ * ), where φ * : V * → V * is the adjoint linear transformation of the dual space V * of V .…”

On Adjoint Entropy of Abelian Groups

“…It is an easy exercise to show that a non-divisible Abelian group G always has subgroups of finite index; indeed in 'most situations' the number of such subgroups is large, being equal to 2 |G| . It is also worth remarking that one can approach this dual entropy by working with the algebraic entropy of the adjoint mapping arising from the theory of Pontryagin duality -such an approach has been exploited in [6] and there is, of course, some overlap between the present work and that paper; although one can often work directly with subgroups of finite index, we have found it convenient, particularly in the early part of Section 2, to quote without proof, results from that work.…”