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

Christopher Good, R Knight, B Raines

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


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 Henon 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.
Original languageEnglish
Pages (from-to)267-289
Number of pages23
JournalFundamenta Mathematicae
Issue number3
Early online date1 Jan 2006
Publication statusPublished - 1 Jan 2006


  • continuum
  • indecomposable
  • inverse limits
  • unimodal
  • attractor
  • invariant set


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