Problem 114

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$det (I - A) $ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?


This may be related to Theorem 5.3 of [1] and the remark immediately following it.  That result says that for a one-dimensional Axiom A basic set $\Lambda$ with filtration pair $(X,E)$ the homology $H_{u+1}(X,E, \mathbb S^1) \cong ker( I-A: T^n \to T^n),$ where $u$ is the dimension of the unstable bundle of $\Lambda$ and $A$ is a signed Markov partition matrix for $\Lambda$.  If $det(I-A) \ne 0$ then it equals the order of the group $ker( I-A: T^n \to T^n) \cong \mathbb Z^n / (I-A) \mathbb Z^n.$

The question would be to what extent similar results hold for higher dimensional basic sets.


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