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

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 ( x g ) = 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 τ 1 , … , τ t is the distinct A -closed orbits that covers τ ˜ . Then, deg ( τ 1 / τ ˜ ) + ⋯ + deg ( τ t / τ ˜ ) = | G | . Now, let l ¯ = ( deg ( τ 1 / τ ˜ ) , … , deg ( τ t / 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 A l ¯ = { 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.