International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 17, Pages 1047-1053

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

Mohd. Salmi Md. Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor Darul Ehsan, Bangi 43600 UKM, Malaysia

Received 21 August 2002

Copyright © 2003 Mohd. Salmi Md. Noorani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let A:TT 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 τ, we have A(xg)=A(x)τ(g) for all xT and gG. Now, let A˜:T˜T˜ be the (ergodic) automorphism induced by the G-action on T. Let τ˜ be an A˜-closed orbit (i.e., periodic orbit) and τ an A-closed orbit which is a lift of τ˜. Then, the degree of τ over τ˜ is defined by the integer deg(τ/τ˜)=λ(τ)/λ(τ˜), where λ() denotes the (least) period of the respective closed orbits. Suppose that τ1,,τt is the distinct A-closed orbits that covers τ˜. Then, deg(τ1/τ˜)++deg(τt/τ˜)=|G|. Now, let l¯=(deg(τ1/τ˜),,deg(τt/τ˜)). 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 τ˜ induces the partition l¯ of the integer |G|. Our aim in this paper is to characterize this partition l¯ for which Al¯={τ˜T˜:τ˜ 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.