Homoclinic points and isomorphism rigidity of algebraic ℤ d -actions on zero-dimensional compact abelian groups

摘要

Letd>1, and letα andβ be mixing ℤ d -actions by automorphisms of zero-dimensional compact abelian groupsX andY, respectively. By analyzing the homoclinic groups of certain sub-actions ofα andβ we prove that, if the restriction ofα to some subgroup Γ ⊂ ℤ d of infinite index is expansive and has completely positive entropy, then every measurable factor mapφ: (X, α)→(Y, β) is almost everywhere equal to an affine map. The hypotheses of this result are automatically satisfied if the actionα contains an expansive automorphismα n ,n ∈ ℤ d , or ifα arises from a nonzero prime ideal in the ring of Laurent polynomials ind variables with coefficients in a finite prime field. Both these corollaries generalize the main theorem in [9]. In several examples we show that this kind of isomorphism rigidity breaks down if our hypotheses are weakened.