Anna Giordano Bruno

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. This resolves in a strong way an open question from [7] .

Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σλ: K Γ → K Γ is defined by (x i ) i∈Γ ↦ (x λ(i)) i∈Γ [3]. In this article we compute the algebraic entropy of σλ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2].

Algebraic entropy of shift endomorphisms on Abelian groups

Abstract For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σλ of the direct sum ⊕Γ K is the endomorphism defined by (x i ) iεΓ ↦ (x λ(i)) iεΓ [3]. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function λ. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.

Pseudocompact totally dense subgroups

It was shown in [1] that a compact abelian group K that admits a proper totally dense pseudocompact subgroup cannot have a bounded torsion closed G -subgroup; moreover this condition was shown to be also sucient for the existence of a proper totally dense pseudocompact subgroup of K under the Lusin’s Hypothesis. We prove in ZFC that this condition ensures actually the existence of a proper totally dense subgroup H of K that contains an !-bounded dense subgroup of K (such an H is necessarily pseudocompact). This answers two questions from [1].

Entropy in a Category

The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. Indeed, we introduce the concept of entropy function h of an abelian category, and we define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion-free class of a torsion theory.

Topological entropy in totally disconnected locally compact groups

Let

About the Algebraic Yuzvinski Formula

G

STRING NUMBERS OF ABELIAN GROUPS

be a topological group, let

STRINGS OF GROUP ENDOMORPHISMS

\phi

Characterized Subgroups of Topological Abelian Groups

be a continuous endomorphism of