Inhomogeneities in non-hyperbolic one-dimensional invariant sets

Brian E. Raines¹

¹ Department of Mathematics Baylor University Waco, TX 76798-7328, U.S.A. and Mathematical Institute University of Oxford Oxford OX1 3LB, UK

Fundamenta Mathematicae, Tome 182 (2004) no. 3, pp. 241-268

Résumé

The topology of one-dimensional invariant sets (attractors) is of great interest. R. F. Williams [20] demonstrated that hyperbolic one-dimensional non-wandering sets can be represented as inverse limits of graphs with bonding maps that satisfy certain strong dynamical properties. These spaces have “homogeneous neighborhoods” in the sense that small open sets are homeomorphic to the product of a Cantor set and an arc. In this paper we examine inverse limits of graphs with more complicated bonding maps. This allows us to understand the topology of a wider class of spaces that includes both hyperbolic and non-hyperbolic attractors. Many of these spaces have the property that most small open sets are homeomorphic to the product of a Cantor set and an arc. The interesting “inhomogeneities” occur away from these neighborhoods. By examining the dynamics of the bonding maps that generate these spaces, we characterize the inhomogeneities, and we show that there is a natural nested hierarchy in the collection of these points that is topological.

Détail

DOI : 10.4064/fm182-3-4

Keywords: topology one dimensional invariant sets attractors great interest williams demonstrated hyperbolic one dimensional non wandering sets represented inverse limits graphs bonding maps satisfy certain strong dynamical properties these spaces have homogeneous neighborhoods sense small sets homeomorphic product cantor set arc paper examine inverse limits graphs complicated bonding maps allows understand topology wider class spaces includes hyperbolic non hyperbolic attractors many these spaces have property small sets homeomorphic product cantor set arc interesting inhomogeneities occur away these neighborhoods examining dynamics bonding maps generate these spaces characterize inhomogeneities there natural nested hierarchy collection these points topological