### 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 language | English |
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Pages (from-to) | 1047-1053 |

Number of pages | 7 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 2003 |

Issue number | 17 |

DOIs | |

Publication status | Published - 2003 |

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### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

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

Research output: Contribution to journal › Article

*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 17, pp. 1047-1053. https://doi.org/10.1155/S0161171203208164

}

TY - JOUR

T1 - Closed orbits of (G,T)-extension of ergodic toral automorphisms

AU - Md. Noorani, Mohd. Salmi

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=17844391520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17844391520&partnerID=8YFLogxK

U2 - 10.1155/S0161171203208164

DO - 10.1155/S0161171203208164

M3 - Article

AN - SCOPUS:17844391520

VL - 2003

SP - 1047

EP - 1053

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 17

ER -