On Fixed Points of Knaster On Fixed Points of Knaster ContinuaContinua

Vincent A SsembatyaVincent A SsembatyaMakerere University UgandaMakerere University Uganda

Joint work with James Keesling – University of Florida USAJoint work with James Keesling – University of Florida USA

ContinuaContinua

A continuum is a compact connected A continuum is a compact connected metric space. metric space.

A subcontinuum Y of the continuum A subcontinuum Y of the continuum X is a closed, connected subset of X.X is a closed, connected subset of X.

A composant Com(x) of a given point A composant Com(x) of a given point x in X is the union of all proper x in X is the union of all proper subcontinua in X that contain the subcontinua in X that contain the point x. point x.

Continua contiuedContinua contiued

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

The Inverse limitThe Inverse limit

We give this the relevatised product We give this the relevatised product topology.topology.

The MetricThe Metric

The Solenoid and Knaster The Solenoid and Knaster ContinuaContinua

A Solenoid is a continuum that can be visualized A Solenoid is a continuum that can be visualized as an intersection of a nested sequence of as an intersection of a nested sequence of progressively thinner solid tori that are each progressively thinner solid tori that are each wrapped into the previous one a number of times. wrapped into the previous one a number of times.

Any radial cross-section of a solenoid is a Cantor Any radial cross-section of a solenoid is a Cantor set each point of which belongs to a densely set each point of which belongs to a densely immersed line, called a composant. The immersed line, called a composant. The wrapping numbers may vary from one torus to wrapping numbers may vary from one torus to another; We shall record their sequence by P and another; We shall record their sequence by P and call the associated solenoid the P-adic solenoid.call the associated solenoid the P-adic solenoid.

Two tori – one wrapped in Two tori – one wrapped in another 3 timesanother 3 times

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

Knaster ContinuaKnaster Continua

Diadic Knaster ContinuumDiadic Knaster Continuum

Stage 1Stage 1

Stage 2Stage 2

Stage 3Stage 3

Indecomposable continuaIndecomposable continua

The first indecomposable continuum was The first indecomposable continuum was discovered in 1910 by L E J Brouwer as discovered in 1910 by L E J Brouwer as counterexample to a conjecture of counterexample to a conjecture of Schoenflies that the common boundary Schoenflies that the common boundary between two open, connected, disjoint between two open, connected, disjoint sets in the plane had to be sets in the plane had to be decomposable;decomposable;

Between 1912 and 1920 Janiszewski Between 1912 and 1920 Janiszewski produced more examples of such produced more examples of such continuacontinua

He produced an example in the plane He produced an example in the plane that does not separate the plane;that does not separate the plane;

B. Knaster later gave a simpler B. Knaster later gave a simpler description of this example using description of this example using semicircles – popularly known as the semicircles – popularly known as the Knaster Bucket Handle.Knaster Bucket Handle.

Lots of examples can now be Lots of examples can now be constructed using inverse limit constructed using inverse limit spaces.spaces.

On the fixed point property of On the fixed point property of Knaster ContinuaKnaster Continua

W S Mahavier asked whether everyW S Mahavier asked whether every

homeomorphism of the bucket handle homeomorphism of the bucket handle has at least two fixed points (Continua has at least two fixed points (Continua with the Houston Problem book, p with the Houston Problem book, p 384, Problem 120) - 1979. 384, Problem 120) - 1979.

In response to this question, Aarts and In response to this question, Aarts and Fokkink proved the following theorem Fokkink proved the following theorem in 1998:in 1998:

A homeomorphism of the bucket handle A homeomorphism of the bucket handle (K(2,2,…)) has at least two fixed points.(K(2,2,…)) has at least two fixed points.

They suggested that, in general, for a They suggested that, in general, for a given prime p and any self given prime p and any self homeomorphism g on K(p,p,…), the homeomorphism g on K(p,p,…), the number of elements fixed by the nth number of elements fixed by the nth iterate of g is at least pn.iterate of g is at least pn.

In 2001, we showed their claim to be In 2001, we showed their claim to be false.false.

Main ResultsMain Results

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

For any prime p and any homeomorphism For any prime p and any homeomorphism h on K(p,p, …), h on K(p,p, …),

Some notationSome notation

Other resultsOther results

More resultsMore results

Basis for proofBasis for proof

We remark that our results depend on We remark that our results depend on the fact that isotopies of the Knaster the fact that isotopies of the Knaster continua can be lifted to isotopies of the continua can be lifted to isotopies of the covering solenoid.covering solenoid.

Solenoids are inverse limits of the unit Solenoids are inverse limits of the unit circles;circles;

Knaster continua can be obtained as Knaster continua can be obtained as appropriate quotients with induced maps appropriate quotients with induced maps as Chebychev polynomials on the unit as Chebychev polynomials on the unit intervals;intervals;

Chec CohomologyChec Cohomology

Us Partnerships Us Partnerships vs vs We and TheyWe and They• Each quality outcome can be achieved Each quality outcome can be achieved

only through collaboration.only through collaboration.• Working example: WELCOMINGWorking example: WELCOMING• Working example: IDENTIFICATION OF Working example: IDENTIFICATION OF

THE POPULATIONTHE POPULATION

Basis for proofsBasis for proofs

Other DirectionsOther Directions

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

EndEnd

GeneologyGeneology