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

Mohd Salmi Md Noorani

doi:10.1155/s0161171203208164

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 τ, we have A(xg) = A(x)τ(g) for all x ∈ T and g ∈ G. Now, let be the (ergodic) automorphism induced by the G‐action on T. Let be an ‐closed orbit (i.e., periodic orbit) and τ an A‐closed orbit which is a lift of . Then, the degree of τ over is defined by the integer , where λ( ) denotes the (least) period of the respective closed orbits. Suppose that τ1, …, τt is the distinct A‐closed orbits that covers . Then, . Now, let . Then, the previous equation implies that the t‐tuple is a partition of the integer |G| (after reordering if needed). In this case, we say that induces the partition of the integer |G|. Our aim in this paper is to characterize this partition for which induces the partition is nonempty and provides an asymptotic formula involving the closed orbits in such a set as their period goes to infinity.

Highlights

In a joint paper [3], we studied how periodic orbits of a shift of finite type X lift to a so-called homogeneous extension of X
Our aim in this paper is to extend this latter result to ergodic toral automorphisms
The first step is to bring in a socalled cyclic extension and obtain the associated Chebotarev theorem for this extension

Summary

Introduction

In a joint paper [3], we studied how periodic orbits of a shift of finite type X lift to a so-called homogeneous extension of X. We consider a (G, τ)-extension of an ergodic toral automorphism Aand provide an analogous result to the one obtained in Noorani and Parry [3]. These identifications are used to obtain an auxiliary result, the so-called Chebotarev theorem for the G-extension of A (see Parry and Pollicott [4]). Recall that Ais the ergodic toral automorphism induced by the G-action on A.

Results

Conclusion