On Fixed Points of Knaster Continua

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On Fixed Points of Knaster Continua

• Vincent A Ssembatya
• Makerere University Uganda
• Joint work with James Keesling University of
  Florida USA

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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.

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Continua contiued

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

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The Inverse limit

• We give this the relevatised product topology.

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The Metric
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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.

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Two tori one wrapped in another 3 times
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An approximation of (3,2,2) solenoid.
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Knaster Continua
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Diadic Knaster Continuum
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Stage 1
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Stage 2
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Stage 3
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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

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• 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.

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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

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• 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.

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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,
  ),

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Some notation
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Other results
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More results
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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

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Chec Cohomology

• Us Partnerships vs We and They
• Each quality outcome can be achieved only through
  collaboration.
• Working example WELCOMING
• Working example IDENTIFICATION OF THE
  POPULATION

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Basis for proofs
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Other Directions

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

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• End

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Geneology