Pisot-Vijayaraghavan numbers and positive matrices. (English) Zbl 0989.57500

Williams, R. F.

Aoki, N. (ed.) et al., Dynamical systems and chaos. Vol. 1: Mathematics, engineering and economics. Proceedings of the international conference, Hachioji, Japan, May 23-27, 1994. Singapore: World Scientific. 268-277 (1995).

Summary: Branched manifolds were introduced in 1969. (The 1-dimensional case was introduced in 1967 and was used to study Anosov diffeomorphisms in 1968; the latter was popularized under the name ‘train track’ somewhat later.) Branched manifolds have tangent bundles, so that such concepts as orientability and immerison make sense. However, the top-dimensional homology of our branched manifolds is 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 are able to prove that ‘weak shift equivalence’ implies ‘strong shift equivalence’ for certain matrices. Though our theory works, at least in principle, in all dimensions, we carry out the proofs only in the 3-dimensional case.
For the entire collection see [Zbl 0990.00505].

MSC:

57M99	General low-dimensional topology
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure
57N10	Topology of general \(3\)-manifolds (MSC2010)
37-XX	Dynamical systems and ergodic theory