Itineraries of maps of dendrites and dendroids having one "turning point"
by
Stewart Baldwin
Auburn University

If f is a map on a closed interval I having one turning point t, then it is well known that the dynamics of f is essentially determined by the itinerary of t (i.e., the sequence of components of I \{t} visited by the orbit of t). One way of stating this would be as follows. Theorem: If f (resp. g) is a map on the interval D (resp. E) having exactly one turning point t (resp. u), no two points of D (resp. E) have the same itinerary with respect to f (resp. g), and no proper subinterval of D (resp. E) contains the orbit of t (resp. u), then f and g are conjugate iff t and u have the ßame" itinerary.

Furthermore, given any interval map with exactly one turning point, it can be obtained from a map satisfying the hypothesis of the above theorem by "dynamically trivial" modifications (enlarging the interval at the ends, or blowing points up to intervals).

As it turns out, the above well known results are also true for dendrites. In fact, the words interval and subinterval can simply be replaced by dendrite and subdendrite in the theorem above, where a turning point in this case is defined as a point where the function is not locally one-to-one. Questions that will be discussed include:

1. To what extent can the structure of D and f be reconstructed from the sequence describing the itinerary of t?

2. What kinds of sequences can be realized as the itinerary of the turning point of some dendrite map satisfying the above hypotheses?

3. To what extent can topological properties of f and D be characterized by properties of the itinerary?

If time permits, the more complicated situation for dendroids (not necessarily locally connected) will be briefly discussed.

Date received: February 28, 2003

Copyright © 2003 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 # cajz-93.