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
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.
