On Fixed Points of Knaster Continua

1. On Fixed Points of Knaster Continua Vincent A Ssembatya Makerere University Uganda Joint work with James Keesling – University of Florida USA

2. Continua • A continuum is a compact connected metric space. • A subcontinuum Y of the continuum X is a closed, connected subset of X. • A composant Com(x) of a given point x in X is the union of all proper subcontinua in X that contain the point x.

3. Continua contiued A continuum is indecomposable if it is not the union of two of its proper nonempty subcontinua.

4. The Inverse limit We give this the relevatised product topology.

5. The Metric

6. The Solenoid and Knaster Continua A Solenoid is a continuum that can be visualized as an intersection of a nested sequence of progressively thinner solid tori that are each wrapped into the previous one a number of times. Any radial cross-section of a solenoid is a Cantor set each point of which belongs to a densely immersed line, called a composant. The wrapping numbers may vary from one torus to another; We shall record their sequence by P and call the associated solenoid the P-adic solenoid.

7. Two tori – one wrapped in another 3 times

8. An approximation of (3,2,2…) solenoid.

9. Knaster Continua

10. DiadicKnaster Continuum

11. Stage 1

12. Stage 2

13. Stage 3

14. Indecomposable continua The first indecomposable continuum was discovered in 1910 by L E J Brouwer as counterexample to a conjecture of Schoenflies that the common boundary between two open, connected, disjoint sets in the plane had to be decomposable; Between 1912 and 1920 Janiszewski produced more examples of such continua

15. He produced an example in the plane that does not separate the plane; B. Knaster later gave a simpler description of this example using semicircles – popularly known as the Knaster Bucket Handle. Lots of examples can now be constructed using inverse limit spaces.

16. On the fixed point property of Knaster Continua • W S Mahavier asked whether every homeomorphism of the bucket handle has at least two fixed points (Continua with the Houston Problem book, p 384, Problem 120) - 1979. • In response to this question, Aarts and Fokkink proved the following theorem in 1998:

17. A homeomorphism of the bucket handle (K(2,2,…)) has at least two fixed points. • They suggested that, in general, for a given prime p and any self homeomorphism g on K(p,p,…), the number of elements fixed by the nth iterate of g is at least pn. • In 2001, we showed their claim to be false.

18. Main Results For any old prime p, there is a homeomorphism g on K(p,p,…) with a single fixed point. For any prime p and any homeomorphism h on K(p,p, …),

19. Some notation

20. Other results

21. More results

22. Basis for proof • We remark that our results depend on the fact that isotopies of the Knaster continua can be lifted to isotopies of the covering solenoid. • Solenoids are inverse limits of the unit circles; • Knaster continua can be obtained as appropriate quotients with induced maps as Chebychev polynomials on the unit intervals;

23. ChecCohomology • Us Partnerships vsWe and They • Each quality outcome can be achieved only through collaboration. • Working example: WELCOMING • Working example: IDENTIFICATION OF THE POPULATION

24. Basis for proofs

25. Other Directions We have constructed higher dimensional Knaster Continua and Proved isotopy lifting properties for these (except in dimension 2).

26. End

27. Geneology