A new example of a tree-like continuum
with a fixed-point-free self-map
L. C. Hoehn ([email protected])
joint with Rodrigo Hernández-Gutiérrez
Nipissing University
August 4, 2016
31st Summer Conference on Topology and its Applications
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
1/7
Fixed-point-free maps on tree-like continua
Continuum ≡ compact, connected, metric space
X is tree-like if X ≈ limhTn , fn i, where each Tn is a tree.
←
−
g : X → X is fixed-point-free if g (x) 6= x for all x ∈ X .
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
2/7
Fixed-point-free maps on tree-like continua
Continuum ≡ compact, connected, metric space
X is tree-like if X ≈ limhTn , fn i, where each Tn is a tree.
←
−
g : X → X is fixed-point-free if g (x) 6= x for all x ∈ X .
Bellamy 1980: There is a tree-like continuum X with a fixed-point-free
map g : X → X .
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
2/7
Fixed-point-free maps on tree-like continua
Continuum ≡ compact, connected, metric space
X is tree-like if X ≈ limhTn , fn i, where each Tn is a tree.
←
−
g : X → X is fixed-point-free if g (x) 6= x for all x ∈ X .
Bellamy 1980: There is a tree-like continuum X with a fixed-point-free
map g : X → X .
Fearnley & Wright 1993: Geometric description of example in R3 .
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
2/7
Fixed-point-free maps on tree-like continua
Continuum ≡ compact, connected, metric space
X is tree-like if X ≈ limhTn , fn i, where each Tn is a tree.
←
−
g : X → X is fixed-point-free if g (x) 6= x for all x ∈ X .
Bellamy 1980: There is a tree-like continuum X with a fixed-point-free
map g : X → X .
Fearnley & Wright 1993: Geometric description of example in R3 .
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
2/7
Fixed-point-free maps on tree-like continua
Continuum ≡ compact, connected, metric space
X is tree-like if X ≈ limhTn , fn i, where each Tn is a tree.
←
−
g : X → X is fixed-point-free if g (x) 6= x for all x ∈ X .
Bellamy 1980: There is a tree-like continuum X with a fixed-point-free
map g : X → X .
Fearnley & Wright 1993: Geometric description of example in R3 .
Question
Does there exist a tree-like continuum in the plane R2 with a
fixed-point-free self-map?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
2/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
3/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
Mioduszewski 1963: Each g : X → X is induced by a sequence of
mappings gk : Tnk → Tk
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
3/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
Mioduszewski 1963: Each g : X → X is induced by a sequence of
(after reindexing)
mappings gn : Tn+1 → Tn
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
3/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
Mioduszewski 1963: Each g : X → X is induced by a sequence of
(after reindexing)
mappings gn : Tn+1 → Tn
Oversteegen & Rogers 1980/82: An example with commuting diagrams
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
3/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
Mioduszewski 1963: Each g : X → X is induced by a sequence of
(after reindexing)
mappings gn : Tn+1 → Tn
Oversteegen & Rogers 1980/82: An example with commuting diagrams
Given x = hx1 , x2 , x3 , . . .i ∈ X ,
L. C. Hoehn ([email protected]) (NU)
g (x) = hg1 (x2 ), g2 (x3 ), . . .i ∈ X
Tree-like fixed-point-free
Summer Topology 2016
3/7
Map on inverse limit
Let X = limhTn , fn i.
←
−
Mioduszewski 1963: Each g : X → X is induced by a sequence of
(after reindexing)
mappings gn : Tn+1 → Tn
Oversteegen & Rogers 1980/82: An example with commuting diagrams
Given x = hx1 , x2 , x3 , . . .i ∈ X ,
g (x) = hg1 (x2 ), g2 (x3 ), . . .i ∈ X
f1 (t) 6= g1 (t) for all t ∈ T2 ⇒ g is fixed-point-free
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
3/7
Comparison
Oversteegen &
Rogers example
New example
# Branch points in Tn
3, 9, 33, 129, . . .
2, 4, 7, 13, 25, . . .
Largest degree of
branch point in Tn
18, 30, 42, 54, . . .
5 for all n
# Endpoints in Tn
24, 54, 138, 438, . . .
7, 12, 21, 39, 75, . . .
Valence of fn
24, 96, 384, 1536, . . .
6 for all n
Valence of gn
48, 192, 768, 3072, . . .
12 for all n
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
4/7
New example: continuum
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
fn ∼ 3-fold tent map from spine of Tn+1 to spine of Tn
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
fn ∼ 3-fold tent map from spine of Tn+1 to spine of Tn
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
fn ∼ 3-fold tent map from spine of Tn+1 to spine of Tn
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: continuum
fn ∼ 3-fold tent map from spine of Tn+1 to spine of Tn
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
5/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
New example: fixed-point-free map
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
6/7
Questions
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree? the simple triod?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree? the simple triod?
Question
Does there exist a pair of disjoint commuting maps on a tree?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree? the simple triod?
Question
Does there exist a pair of disjoint commuting maps on a tree? on the
simple triod?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree? the simple triod?
Question
Does there exist a pair of disjoint commuting maps on a tree? on the
simple triod? on a dendrite?
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7
Questions
Question
Is there a coincidence-point-free commuting system
such that. . .
. . . the number of branch points in the trees Tn is bounded?
. . . each Tn is equal to the same tree? the simple triod?
Question
Does there exist a pair of disjoint commuting maps on a tree? on the
simple triod? on a dendrite?
Thank you!
L. C. Hoehn ([email protected]) (NU)
Tree-like fixed-point-free
Summer Topology 2016
7/7