Glasnik Matematicki, Vol. 42, No.1 (2007), 43-56.

STRUCTURE OF INVERSE LIMIT SPACES OF TENT MAPS WITH NONRECURRENT CRITICAL POINTS

Brian Raines and Sonja Štimac

Graduate School of Economics and Business, University of Zagreb, Kennedyev trg 6, 10000 Zagreb, Croatia
e-mail: sonja@math.hr

Abstract.   In this paper we examine the structure of composants of inverse limit spaces generated by tent maps with a nonrecurrent critical point. We identify important structures and substructures of certain composants, and we prove the surprising result that, assuming the critical point is nonrecurrent, there are only finitely many "types" of structures in these composants. This is an important first step towards classifying this family of inverse limit spaces which would in turn lead us closer to a proof of the Ingram Conjecture.

2000 Mathematics Subject Classification.   37B10, 37B45.

Key words and phrases.   Nonrecurrent critical point, tent map, inverse limit, composant, folding point, folding pattern.

DOI: 10.3336/gm.42.1.03

References:

1. M. Barge, K. Brucks and B. Diamond, Self-similarity in inverse limit spaces of tent family, Proc. Amer. Math. Soc. 124 (1996), 3563-3570.
   MathSciNet     CrossRef

2. L. Block, S. Jakimovik, L. Kailhofer and J. Keesling, On the classification of inverse limits of tent maps, Fund. Math. 187 (2005), 171-192.
   MathSciNet

3. K. M. Brucks and B. Diamond, A symbolic representation of inverse limit spaces for a class of unimodal maps, Lecture Notes in Pure and Appl. Math. 170, Dekker, New York, 1995, 207-226.
   MathSciNet

4. H. Bruin, Inverse limit spaces of post-critically finite tent maps, Fund. Math. 165 (2000), 125-138.
   MathSciNet

5. P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics 1, Birkhäuser, Basel, 1980.
   MathSciNet

6. C. Good, R. Knight and B. E. Raines, Non-hyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, preprint (2006)

7. L. Kailhofer, A partial classification of inverse limit spaces of tent maps with periodic critical points, Topology Appl. 123 (2002), 235-265.
   MathSciNet     CrossRef

8. L. Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math. 177 (2003), 95-120.
   MathSciNet

9. B. E. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math. 182 (2004), 241-268.
   MathSciNet

10. S. Stimac, Topological classification of Knaster continua with finitely many endpoints, Ph. D. Thesis (in Croatian, University of Zagreb, 2002.

11. S. Stimac, Structure of inverse limit spaces of tent maps with finite critical orbit, Fund. Math. 191 (2006), 125-150.
    MathSciNet

12. S. Stimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl. 154 (2007), 2265-2281.
    CrossRef