We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.

Entropy on abelian groups

DIKRANJAN, Dikran;GIORDANO BRUNO, Anna
2016-01-01

Abstract

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.
File in questo prodotto:
File Dimensione Formato  
halg.pdf

accesso aperto

Descrizione: Articolo principale, Accesso Aperto MIUR
Tipologia: Versione Editoriale (PDF)
Licenza: Creative commons
Dimensione 731.01 kB
Formato Adobe PDF
731.01 kB Adobe PDF Visualizza/Apri
schedaprogetto.pdf

accesso aperto

Descrizione: Scheda di progetto
Tipologia: Altro materiale allegato
Licenza: Creative commons
Dimensione 49.52 kB
Formato Adobe PDF
49.52 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/1091764
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 38
  • ???jsp.display-item.citation.isi??? 28
social impact