Automorphisms of solenoids and p-adic entropy

We show that a full solenoid is locally the product of a euclidean component and />-adic components for each rational prime p. An automorphism of a solenoid preserves these components, and its topological entropy is shown to be the sum of the euclidean and p-adic contributions. The p-adic entropy of the corresponding rational matrix is computed using its p-adic eigenvalues, and this is used to recover Yuzvinskii's calculation of entropy for solenoidal automorphisms. The proofs apply Bowen's investigation of entropy for uniformly continuous transformations to linear maps over the adele ring of the rationals. 1. Background and results A solenoid is a finite-dimensional, connected, compact abelian group. Equivalently, its dual group is a finite rank, torsion-free, discrete abelian group, i.e. a subgroup of Q for some d > 1. Solenoids generalize the familiar torus groups. Halmos [H] first observed that (continuous) automorphisms of compact groups must preserve Haar measure, providing an interesting class of examples for ergodic theory. Furthermore, Berg [Be] has shown that the entropy of such an automorphism with respect to Haar measure coincides with its topological entropy. We are concerned here with the computation of the topological entropy of an automorphism of a solenoid. If A is such an automorphism, its dual automorphism extends to an element of GL(d,Q), which we also call A (see § 3). When A is a toral automorphism, so A e GL(d, Z), then the topological entropy of A is given by the familiar formula h(A)= I log |A,|, (1) |A,-|>1 where A has complex eigenvalues A, , . . . , Ad counted with multiplicity. To state the generalization to solenoids, let XA(O be the characteristic polynomial of Ae GL(d, Q), and s denote the least common multiple of the denominators of the coefficients of XAU)Yuzvinskii [Y] proved that logs+ I log|A,|. (2) | | * The authors gratefully acknowledge support, respectively, by NSF Grant DMS-8320356 and SERC Award B85318868. 412 D. A. hind and T. Ward Our purpose here is to explain Yuzvinskii's calculation in terms of a combination of geometric and arithmetic hyperbolicity. We begin in § 3 by lifting A to an automorphism of the full solenoid 1. = Q with the same entropy. In Lemma 4.1 we show that the full solenoid is locally a product of a euclidean component and p-adic components for each rational prime p. The entropy of A is computed in Theorem 1 to be the sum of a contribution from the euclidean component, generated by geometric expansion, and contributions from each of the p-adic components, generated by arithmetic expansions. If Qp denotes the p-adic completion of the rationals, than a p-adic component contributes the Bowen entropy h{A; Q) of the uniformly continuous linear map A on the non-compact metric space Qp. Since the infinite place oo on Q> gives the completion ©«, = U, and the Bowen entropy of a linear map on U is given by (1), we can summarize Theorem 1 by h(A)= I h(A;Qp), (3)