On Fixed Points of Knaster Continua. Vincent A Ssembatya Makerere University Uganda Joint work with James Keesling – University of Florida USA. Continua. A continuum is a compact connected metric space. A subcontinuum Y of the continuum X is a closed, connected subset of X.
Vincent A Ssembatya
Makerere University Uganda
Joint work with James Keesling – University of Florida USA
A continuum is indecomposable if it is not the union of two of its proper nonempty subcontinua.
We give this the relevatised product topology.
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.
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
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.
homeomorphism of the bucket handle has at least two fixed points (Continua with the Houston Problem book, p 384, Problem 120) - 1979.
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, …),
We have constructed higher dimensional Knaster Continua and Proved isotopy lifting properties for these (except in dimension 2).