Spaces that won’t say no. (English) Zbl 0876.57044

Williams, R. F.

Ledrappier, François (ed.) et al., 1st international conference on dynamical systems, Montevideo, Uruguay, 1995 - a tribute to Ricardo Mañé. Proceedings. Harlow: Longman. Pitman Res. Notes Math. Ser. 362, 236-246 (1996).

Summary: A class oaf spaces \(K\) and their natural maps \(f: K\to K'\) were introduced in [the author, Publ. Math., Inst. Hautes Étud. Sci. 43(1973), 169-203 (1974; Zbl 0279.58013)] which satisfies the defining property that \(f_*: H_n(K)\to H_n(K')\) is always a “positive matrix”. Here \(n\) is the dimension of both \(K\) and \(K'\); we follow the usage in symbolic dynamics and say that a matrix is positive provided all its entries are \(\geq 0\). The purpose of this paper is to show how a natural construction, related to the “DA” maps of Smale [the author, Global Analysis, Proc. Sympos. Pure Math. 14, 329-334 (1970; Zbl 0213.50303)] often leads to SOB’s, and in particular shows that they exist in all dimensions. Here we present the easier conceptual parts of this theory and elicit a geometric property, which when it holds, leads to quite satisfactory proofs.
For the entire collection see [Zbl 0856.00028].

MSC:

57R35	Differentiable mappings in differential topology
37E99	Low-dimensional dynamical systems

Keywords:

selfmap; homology; symbolic dynamics

Citations:

Zbl 0279.58013; Zbl 0213.50303