Oriented local entropies for expansive actions by commuting automorphisms

Given an expansive action α of ?² by automorphisms of a compact connected metrizable abelian groupX, we show how the entropy of the action may be decomposed into local contributions, 1 $$h(\alpha ) = \sum\limits_{p \leqslant \infty } {h_p^{(a,b)} } (\alpha )$$ in which the summandh _p ^(a,b) (α) represents thep-adic entropy due to arithmetic or geometric hyperbolicity in the direction (a, b). We recognize thep-adic contribution as an integral over thep-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.