A Chebotarev Theorem for finite homogeneous extensions of shifts

Abstract

We derive a Chebotarev Theorem for finite homogeneous extensions of shifts of finite type. These extensions are of the form$\tilde \sigma$:X×G/H→X×G/H where$\tilde \sigma$(x,gH)=(σx, α(x)gH), for some finite groupG and subgroupH. Given a σ-closed orbit τ, the periods of the$\tilde \sigma$-closed orbits covering τ define a partition of the integer |G/H|. The theorem then gives us an asymptotic formula for the number of closed orbits with respect to the various partitions of the integer |G/H|. We apply our theorem to the case of a finite extension and of an automorphism extension of shifts of finite type. We also give a further application to ‘automorphism extensions’ of hyperbolic toral automorphisms.