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Geometric Topology II
September 29 - October 5, 2002
Inter-University Center, Dubrovnik; Department of Mathematics, University of Zagreb
Dubrovnik, Croatia

Organizers
Ivan Ivansic, University of Zagreb;, James E. Keesling, University of Florida;, Alexander N. Dranishnikov, University of Florida;, Sime Ungar, University of Zagreb

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Topological classification of Knaster continua with finitely many endpoints
by
Sonja Stimac
Faculty of Economics, University of Zagreb

In this work we develop a symbolic dynamics method which enables us to study properties of certain classes of inverse limits. We first consider the family of Knaster continua Ks = lim <-- {[0, 1], fs}, where fs : [0, 1] --> [0, 1] are tent functions with slope s in [\surd2, 2] and periodic extreme points. Continua of this family are represented as quotient spaces of two-sided admissible sequences of zeros and ones, with respect to a suitable equivalence relation. We are interested in the structure of the composant of the endpoint [`c] related to the kneading sequence of fs. We define p-i-points characterized by the equivalence relation on the quotient space, and p-bridges, i.e.  specially chosen arcs connecting certain p-i-points. We show that the first (p-1)-bridge in the structure of every p-bridge is of the same type as the first bridge at an arbitrary level which contains the endpoint [`c]. We also show that if there exist two homeomorphic continua in the class we study, then there exists a mapping hq, p between composants of the endpoints and there exists an r in N, r >= p, for which the mapping hq, p maps the first bridge at level q+1 onto the first bridge at level r. From this fact we conclude that the kneading sequences of the corresponding tent functions are equal. In other words, for tent functions fs and ft, s, t in [\surd2, 2], with periodic extreme points, if s =/= t, than the continua Ks and Kt are not homeomorphic.

Bibliography:

  1. Brucks, K. M., Diamond, B., A symbolic representation of inverse limit space for a class of unimodal maps, Continua with the Houston problem book, Eds.: Cook, H. et al., Lecture notes in pure and appl. math. 170, Dekker (1995), 207-226

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

  3. Kailhofer, L., A classification of inverse limit spaces of tent maps with periodic critical points, Preprint (2000)

    4. Date received: August 12, 2002

    Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caje-44.