Deligne periods of mixed motives, 𝐾-theory and the entropy of certain ℤⁿ-actions

Abstract

For a coherent sheaf M on the split n-torus Gn A = spec A[7n] over a commutative ring A there is a natural En-action on its group of global sections F(M) = F(G AA, M). If we give F(M) the discrete topology, we obtain a En-action on the compact abelian Pontrjagin dual F(M)*. The dynamical properties of this action for A = Z have been investigated for some time, especially by K. Schmidt. His book [Sch] gives a comprehensive account of the theory developed so far. In particular, one may ask for the entropy 0 < h(M) < oc of the En-action on F(M)*I which measures to what extent repeated application of the action scatters around points. Note that for any action of Zn on a compact abelian group, topological entropy and metric entropy with respect to Haar measure coincide. The results of Lind, Schmidt and Ward [LSW] (see also [Sch], Chap. V) can be formulated as follows (A = Z): The entropy of M is finite if and only if M belongs to the category T of coherent torsion sheaves on Gn,. By Yuzvinskii's addition formula [Sch], Th. 14.1, entropy defines a homomorphism from the Grothendieck group of T to the reals: h: Ko(T) RI so that it suffices to calculate h(Oz) for irreducible closed subschemes Z C ?n with Z G n. If Z is not defined by a Laurent polynomial, then h(Oz) = 0. On the other hand, for any