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

For a coherent sheaf M on the split n-torus Gn A = spec A7n 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