Addendum: An analogue of Artin reciprocity for closed orbits of skew products

One of the unfulfilled aims of the authors of the preceding paper [W. Parry and M. Pollicott. An analogue of Bauer’s theorem for closed orbits of skew products. Ergod. Th. & Dynam. Sys. 28 (2008), 535–546] was to find a dynamical analogue of Artin reciprocity. In this addendum, we present one such version, suggested by work of Sunada. 0. Introduction In algebraic number theory, one of the most important themes is that of reciprocity. Historically, this originated with the famous quadratic reciprocity theorem, conjectured by Euler and proved by Gauss. This relates the existence of solutions to quadratic equations x2 = p (mod q) to the existence of solutions to quadratic equations x2 = q (mod p). Various extensions and generalizations culminated in Artin’s reciprocity theorem, proved in 1927 [2], from which quadratic reciprocity can be deduced (albeit after some effort). This is a key result in class field theory. Let K be the base field and let L = K (α) be a finite extension with Galois group Gal(L/K ). The Artin reciprocity theorem describes the Galois group Gal(L/K ) in terms of properties of the base field K . More precisely, it gives an isomorphism I/PN ∼= Gal(L/K ). Here I are fractional ideals coprime to an admissible cycle C in L; P are the associated principal fractional ideals; and N is the group of fractional ideals of the form N (B), where B is a fractional ideal which is prime to C (and N denotes the norm). We refer the reader to [1] for details. In this addendum to [4] we present analogous results for subshifts of finite type and hyperbolic flows where closed orbits play the role of prime ideals. In this context, the use of homology leads to simplifications not available in the original number theoretical context.