Closed orbits of (G,T)-extension of ergodic toral automorphisms

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Abstract

Let A:T→T be an ergodic automorphism of a finite-dimensional torus T. Also, let G be the set of elements in T with some fixed finite order. Then, G acts on the right of T, and by denoting the restriction of A to G by T, we have A ( xg)=A (X)T (g) for all x∈T and g∈G. Now, let A: T→T be the (ergodic) automorphism induced by the G-action on T. Let T be an A∼-closed orbit (i.e., periodic orbit) and T an A-closed orbit which is a lift of T. Then, the degree of T over T is defined by the integer deg ( T/T)=λ (T)/λ (T), where λ ( ) denotes the (least) period of the respective closed orbits. Suppose that T1,...,Tt is the distinct A-closed orbits that covers T. Then, deg ( T1/T)+□+deg ( Tt/T)= |G|. Now, let l-= ( deg (T1/T),...,deg ( Tt/T)). Then, the previous equation implies that the t-tuple l- is a partition of the integer |G| (after reordering if needed). In this case, we say that T induces the partition l- of the integer |G|. Our aim in this paper is to characterize this partition l- for which Al-= {T⊂T:T induces the partition l-} is nonempty and provides an asymptotic formula involving the closed orbits in such a set as their period goes to infinity.

Original languageEnglish
Pages (from-to)1047-1053
Number of pages7
JournalInternational Journal of Mathematics and Mathematical Sciences
Volume2003
Issue number17
DOIs
Publication statusPublished - 2003

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Closed Orbit
Automorphisms
Partition
Automorphism
Integer
Reordering
Asymptotic Formula
Periodic Orbits
Torus
Infinity
Cover
Denote
Restriction
Distinct
Imply

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Cite this

Closed orbits of (G,T)-extension of ergodic toral automorphisms. / Md. Noorani, Mohd. Salmi.

In: International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 17, 2003, p. 1047-1053.

Research output: Contribution to journalArticle

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