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

Volume 192 / 2006

Chris Good, Robin Knight, Brian Raines Fundamenta Mathematicae 192 (2006), 267-289 MSC: 37B45, 37E05, 54F15, 54H20. DOI: 10.4064/fm192-3-6


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.


  • Chris GoodSchool of Mathematics and Statistics
    University of Birmingham
    Birmingham, B15 2TT, UK
  • Robin KnightMathematical Institute
    University of Oxford
    Oxford OX1 3LB, UK
  • Brian RainesDepartment of Mathematics
    Baylor University
    Waco, TX 76798-7328, U.S.A.

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