Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

Chris Good ¹   ; Robin Knight ²   ; Brian Raines ³

¹ School of Mathematics and Statistics University of Birmingham Birmingham, B15 2TT, UK
² Mathematical Institute University of Oxford Oxford OX1 3LB, UK
³ Department of Mathematics Baylor University Waco, TX 76798-7328, U.S.A.

Fundamenta Mathematicae, Tome 192 (2006) no. 3, pp. 267-289

Résumé

We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or attracting condition. We show that for a family of conceptual models for the Hénon attractor, inverse limits of tent maps, these restrictions characterize the structure of inhomogeneities. We end with several results regarding the collection of parameters that generate such spaces.

Détail

DOI : 10.4064/fm192-3-6

Keywords: examine structure countable closed invariant sets under dynamical system compact metric space motivated desire understand possible structures inhomogeneities one dimensional nonhyperbolic sets inverse limits finite graphs particularly those inhomogeneities form countable set using tools descriptive set theory prove surprising restriction topological structure these invariant sets map satisfies weak repelling attracting condition family conceptual models attractor inverse limits tent maps these restrictions characterize structure inhomogeneities end several results regarding collection parameters generate spaces