Oriented local entropies for expansive actions by commuting automorphisms

Given an expansive action a of Z2 by automorphisms of a compact connected metrizable abelian group X, we show how the entropy of the action may be decomposed into local contributions, h(a) = å p £ ¥ hp(a,b)(a) (1) in which the summand hp(a,b)(a) represents the p-adic entropy due to arithmetic or geometric hyperbolicity in the direction (a,b). We recognize the p-adic contribution as an integral over the p-adic unit circle, in analogy with the global counterpart. As (a,b) changes, the decomposition (1) changes only when the line through (a,b) passes through one of a finite collection of critical directions, which are explicitly identified.