An analogue of Bauer’s theorem for closed orbits of skew products

In this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of hyperbolic systems. 0. Introduction Let φ : X → X be a hyperbolic dynamical system and, for a finite G, let φ̃ : X̃ → X̃ be a G-extension for which φ̃ is also hyperbolic. Given φ we are interested in describing the possibilities for φ̃ in terms of the closed φ-orbits. This is analogous to a classical problem in number theory which asks for a description of all finite Galois extensions of an algebraic number field in terms of its prime ideals. This lies at the heart of class field theory. Let L be a Galois extension of a number field K (i.e. one for which the automorphism group fixing K is transitive). A prime ideal p for K corresponds to an ideal for L , which may be a product of prime ideals from L . We say that p splits if it is a product of distinct prime ideals in L . The following result characterizes a Galois extension in terms of which primes split in it [6]. BAUER’S THEOREM. AGalois extension L of K is uniquely determined by the set of prime ideals that split in it (i.e. if L1, L2 ⊃ K are Galois extensions and an ideal p in K splits in L1 if and only if it splits in L2 then L1 is isomorphic to L2). We can use the analogy of a dynamical system to a number field whereby a G-extension (covering) corresponds to a Galois extension and where closed orbits play the role of primes (cf. [11, 12, 14]). In particular, in this note we want to consider analogues of Bauer’s theorem for dynamical systems. We are interested in how local information on closed prime orbits and their Frobenius elements classifies skew products. † Bill Parry died before this paper was completed. I have attempted to write the paper as we had originally planned. All errors are entirely my responsibility. (M.P.)