Arc and a triod

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Arc and a triod

• I x 0 x 1
• T (r,q) 0 r 1 q e 0,
  p/2, p
• Continuum compact connected metric space

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Inverse Limits

• X
• Y

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Inverse Limits

• X
• Examples
• f(t) t X I define h((ti)) t where ti
  t
• f(t)

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Inverse Limits

• X

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Inverse Limits

• X

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Non-confluent maps

• h I S1 is confluent iff whenever K is a
  subcontinuum
• of S1, and W is any component of h-1K, then
  hW K.

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Non-confluent maps

• h I T is confluent iff whenever K is a
  subcontinuum
• of T, and W is any component of h-1K, then
  hW K.

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Arc-like

• X is arc-like iff for each e gt 0, there is a map
  f X I
• such that diam f -1 (f(x)) lt e for all x in X.
• X (x, sin1/x)0ltxlt2.1U(0,y)-1ltylt1

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Arc-like

• X is arc-like iff for each e gt 0, there is a map
  f X I
• such that diam f -1(f(x)) lt e for all x in X.

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T-like with a triod

• X is T-like iff for each e gt 0, there is a map
• f X T such that diam f -1(f(x)) lt e for
  all
• x in X.

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Confluent maps

• Theorem
• The confluent images of arc-like curves are
  tree-like.
• Conjecture
• The confluent images of arc-like curves are
  arc-like.
• So when is the inverse limit of trees, arc-like?

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6 mapsnot arc-like
arc-like
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Span of a mapSpan of T 1

• Definition - If f T -gt T, then s(f)
• lube conn. H T xT, p1(H) p2(H) and
  d(f(x),f(y))gte
• Definition - If X is a space, then the span of X
  is s(Id).

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Not arc-like

• 1964 A. Lelek proved every arc-like continua has
  span zero which means s(Id) 0.
• Bing asked if there exists an atriodic tree-like
  continuum that is not arc-like.
• R. D. Anderson said yes in an abstract.
• 1972 Ingram proves that the answer to Bings
  question is yes.

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s(f n ) gt ½ gt s (Y) gt ¼

• If f T -gt T, then the span of f
• lube conn. H T xT, p1(H) p2(H) and
  d(f(x),f(y))gte

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K is a subset of YxY gt Y has positive span gtY
is not arc-like
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Y is atriodic

• 1968 Ingram If every proper subcontinuum of Y
  is arc-like, then Y is atriodic.
• Let M be a proper subcontinuum of Y and choose N
  so that if n gt N pnM is not equal to T.
• Case 1 For all i, there exists jgti so that O is
  not in pjM
• thus pjM is an arc and M is
  arc-like
• Case 2 There exists a k such that if j gt k then
  O is in pjM. Choose j gt N and jgtk. Then O is
  in pj3M thus B is in pj2M gt fOB is in
  pj1M gt T is a subset of pjM gtlt

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arc-like
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arc-like
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To see that it is arc-like
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This time T-like was arc-like

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When is T-like arc-like ?

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2003 Ryden
Theorem 15 If f is a light simplicial map from a
connected graph G into a graph, then f can be
factored through an arc iff there is a finite
sequence of folds pi and a simplicial map y
such that y pn pn-1. . . p2 p1 G I

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2003 Ryden