Celebratio Mathematica — Williams — Complete Bibliography

Complete Bibliography

Works connected to Kenichi Shiraiwa

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R. F. Williams: “Pisot–Vijayarghavan numbers and positive matrices,” pp. 268–277 in Dynamical systems and chaos (Hachioji, Japan, 23–27 May 1994), vol. 1. Edited by N. Aoki, K. Shiraiwa, and Y. Takahashi. World Scientific (River Edge, NJ), 1995. MR 1479944 Zbl 0989.57500 incollection

Branched manifolds, were introduced in [1969, 1974]. (The 1-dimensional case was introduced in [1967] and was used to study Anosov diffeomorphisms in [1970]; this last was popularized under the name “train track” [Fathi et al. 1979] somewhat later.) Branched manifolds have tangent bundles, so that such concepts as orientability and immersion make sense. However, the top dimensional homology of our branched manifolds are typically of dimension \( k > 1 \), so that there are many choices of generators, consistent with the chosen orientation. Thus we say that an oriented branched \( n \)-manifold \( K \) is sensed provided the \( n \) dimensional homology is free abelian, and that generators are so chosen that the homomorphisms induced by immersions, relative to the chosen bases, are matrices whose entries are all \( \geq 0 \).(Or \( \leq 0 \). Here the inequalities depend upon whether the immersion preserves or reverses the chosen orientations.) Sensed, oriented, branched manifolds are called “SOB’s” for short. We use this concept to prove a partial converse to the Perron–Frobenius theorem, and expect to be able to prove that “weak shift equivalence” implies “strong shift equivalence” for certain matrices. Though our theory works, at least in principal, in all dimensions, we have carried out the proofs only in the 3 dimensional case.

Yōichirō Takahashi	Related
Nobuo Aoki	Related
Kenichi Shiraiwa	Related