Abstract

Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established.

Highlights

  • Square nonnegative integer matrices are used to describe maps on Cantor sets known as subshifts of finite type

  • For nontrivial irreducible incidence matrices John Franks has shown that flow equivalence is completely determined by two invariants, the Parry-Sullivan number and

  • We only need a means of assigning orientations to rectangles of a Markov partition

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Summary

Flow equivalence

Square nonnegative integer matrices are used to describe maps on Cantor sets known as subshifts of finite type. Two such matrices are flow equivalent if their induced subshifts of finite type give rise to topologically equivalent suspension flows. In terms of the corresponding subshift and suspension, irreducibility is equivalent to the existence of a dense orbit. Irreducible permutation matrices give rise to flows with a single closed orbit and are said to form the trivial flow equivalence class. For nontrivial irreducible incidence matrices John Franks has shown that flow equivalence is completely determined by two invariants, the Parry-Sullivan number and. Zn BF (A) = (I − A)Zn are the Parry-Sullivan number and the Bowen-Franks group respectively. Huang has settled the difficult classification problem arising when the assumption of irreducibility is dropped, [3], [4], [5]

Twist-wise flow equivalence
Matrix moves
The double cover flow
Other Abelian groups
Examples
A B C1 C2 D E

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