zbMATH — the first resource for mathematics

Composants of the horseshoe. (English) Zbl 0818.54028
Summary: The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map is one of the standard examples in continua theory as well as in dynamical systems. It is not arcwise connected. Its arcwise components coincide with composants, and with unstable manifolds in the dynamical setting. Knaster asked whether these composants are all homeomorphic, with the obvious exception of the zero composant. Partial results were obtained by D. P. Bellamy [Houston J. Math. 5, 313-318 (1979; Zbl 0457.54031)], W. Dȩbski and E. D. Tymchatyn [Topology Proc. 12, No. 2, 239-256 (1987; Zbl 0674.54023)], and by J. M. Aarts and R. J. Fokkink [Fundam. Math. 139, No. 3, 193-208 (1991; Zbl 0759.54017)]. We answer Knaster’s question in the affirmative. The main tool is a very simple type of symbolic dynamics for the horseshoe and related continua.

54H20 Topological dynamics (MSC2010)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
54F15 Continua and generalizations
Full Text: DOI EuDML