Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Bratteli Diagrams for Cantor aperiodic
systems:
Construction and Applications
Texas A&M University
Plan
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
1
Bratteli diagrams for Cantor aperiodic systems
2
Application to invariant measures
Introduction
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
A Cantor set X is a 0-dimensional compact metric
space without isolated points, e.g. X = 2Z .
T : X → X is a homeomorphism: (X , T ) is a Cantor
dynamical system. These are shifts, odometers,
substitional systems, etc.
All orbits of T are infinite: (X , T ) is called aperiodic.
Introduction
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
A Cantor set X is a 0-dimensional compact metric
space without isolated points, e.g. X = 2Z .
T : X → X is a homeomorphism: (X , T ) is a Cantor
dynamical system. These are shifts, odometers,
substitional systems, etc.
All orbits of T are infinite: (X , T ) is called aperiodic.
Our goal:
To develop the classification theory for arbitrary Cantor
aperiodic systems.
Motivation
Cantor
Aperiodic
Systems
Bratteli
diagrams
1
Construction of Bratteli diagrams for minimal
homeomorphisms by Herman, Putnam, and Skau:
Ordered Bratteli diagrams, dimension groups and
topological dynamics, Internat. J. Math., 1992.
2
Classification results for Cantor minimal systems
by Giordano, Putnam, and Skau: Topological orbit
equivalence and C ∗ -crossed products, J. Reine Angew.
Math., 1995.
Invariant
measures
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
V0
Bratteli
diagrams
Invariant
measures
E1
V1
E2
V2
E3
V3
A Bratteli diagram B is an
infinite graph (V , E) such
that
s(En ) = Vn−1 ;
r (En ) = Vn+1 ;
r −1 (v ) 6= ∅ ∀v ∈ V \ V0 ;
s−1 (v ) 6= ∅ ∀v ∈ V ;
V0 is a singleton.
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Consider an infinite path.
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider an infinite path.
Invariant
measures
XB is the set of all
infinite paths.
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider an infinite path.
Invariant
measures
XB is the set of all
infinite paths.
Topology: two paths are
close if they agree on a
large initial segment.
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider an infinite path.
Invariant
measures
XB is the set of all
infinite paths.
Topology: two paths are
close if they agree on a
large initial segment.
XB is a Cantor set.
Bratteli diagrams for Cantor aperiodic systems
Concept of unordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider an infinite path.
Invariant
measures
XB is the set of all
infinite paths.
Topology: two paths are
close if they agree on a
large initial segment.
XB is a Cantor set.
How to introduce
dynamics on XB ?
Bratteli diagrams for Cantor aperiodic systems
Concept of ordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Bratteli diagrams for Cantor aperiodic systems
Concept of ordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Consider a vertex
v ∈ V \ V0 .
Bratteli diagrams for Cantor aperiodic systems
Concept of ordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Consider a vertex
v ∈ V \ V0 .
Consider the set r −1 (v ).
Bratteli diagrams for Cantor aperiodic systems
Concept of ordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider a vertex
v ∈ V \ V0 .
Invariant
measures
3
0
21
Consider the set r −1 (v ).
Enumerate edges from
r −1 (v ).
Bratteli diagrams for Cantor aperiodic systems
Concept of ordered Bratteli diagram
Cantor
Aperiodic
Systems
Bratteli
diagrams
Consider a vertex
v ∈ V \ V0 .
Invariant
measures
3
0
21
Consider the set r −1 (v ).
Enumerate edges from
r −1 (v ).
Do the same for every
vertex.
Bratteli diagrams for Cantor aperiodic systems
Maximal and minimal paths
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
0
0
1
1
1
1
0
3
0
3
0
12
0
12
Bratteli diagrams for Cantor aperiodic systems
Maximal and minimal paths
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
0
0
1
1
1
1
0
3
0
3
0
12
0
12
x = (xn ) is maximal if xn
carries the maximal
number amongst
r −1 (r (xn )).
Bratteli diagrams for Cantor aperiodic systems
Maximal and minimal paths
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
0
0
1
1
1
1
0
0
3
3
0
0
12
12
x = (xn ) is maximal if xn
carries the maximal
number amongst
r −1 (r (xn )).
Denote by Xmax and Xmin
the set of all maximal
and minimal paths.
Bratteli diagrams for Cantor aperiodic systems
Maximal and minimal paths
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
0
0
1
1
1
1
0
0
3
3
0
0
12
12
x = (xn ) is maximal if xn
carries the maximal
number amongst
r −1 (r (xn )).
Denote by Xmax and Xmin
the set of all maximal
and minimal paths.
Xmax 6= ∅ and Xmin 6= ∅.
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
0
0
1
1
1
1
0
3
0
3
0
12
0
12
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Define
Bratteli
diagrams
ϕB : XB \ Xmax → XB \ Xmin :
Invariant
measures
0
0
1
1
1
1
0
3
0
3
0
12
0
12
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Define
Bratteli
diagrams
ϕB : XB \ Xmax → XB \ Xmin :
Invariant
measures
0
0
1
1
1
1
0
3
0
3
Fix x ∈ XB \ Xmax .
0
12
0
12
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Define
Bratteli
diagrams
ϕB : XB \ Xmax → XB \ Xmin :
Invariant
measures
0
0
1
1
1
1
0
3
0
3
Fix x ∈ XB \ Xmax .
0
12
0
12
Find the first k with xk
non-maximal.
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Define
Bratteli
diagrams
ϕB : XB \ Xmax → XB \ Xmin :
Invariant
measures
0
1
1
0
3
0
Fix x ∈ XB \ Xmax .
12
Find the first k with xk
non-maximal.
Take the successor of xk .
0
1
1
0
3
0
12
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Define
Bratteli
diagrams
ϕB : XB \ Xmax → XB \ Xmin :
Invariant
measures
0
1
1
0
3
0
Fix x ∈ XB \ Xmax .
12
Find the first k with xk
non-maximal.
Take the successor of xk .
0
1
1
0
3
0
12
Connect the successor to the
top vertex by the minimal
path.
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
ϕB is defined everywhere but Xmax .
Each x ∈ XB has a unique preimage except for
x ∈ Xmin .
Definition
If we can define ϕB on Xmax with the range of Xmin such that
ϕB : XB → XB is a homeomorphism, then (XB , ϕB ) is called
a Bratteli-Vershik system.
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Question:
Does any ordering on the diagram define a continuous
Vershik map?
Answer:
No
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Question:
Does any ordering on the diagram define a continuous
Vershik map?
Answer:
No
If Xmax and Xmin are singletons, then forcing
ϕB (Xmax ) = Xmin we get that ϕB : XB → XB is a
homeomorphism.
Bratteli diagrams for Cantor aperiodic systems
Vershik homeomorphism
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Question:
Does any ordering on the diagram define a continuous
Vershik map?
Answer:
No
If Xmax and Xmin are singletons, then forcing
ϕB (Xmax ) = Xmin we get that ϕB : XB → XB is a
homeomorphism.
Question:
Can any diagram be given an ordering that defines a
continuous Vershik map?
Answer:
No
Bratteli diagrams for Cantor aperiodic systems
Bratteli diagram without continuous dynamics
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Bratteli diagrams for Cantor aperiodic systems
From Cantor aperiodic systems to Bratteli diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Theorem (Herman, Putnam, Skau (1992))
Each Cantor minimal system (X , T ) is conjugate to a
Bratteli-Vershik system (XB , ϕB ).
Bratteli diagrams for Cantor aperiodic systems
From Cantor aperiodic systems to Bratteli diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Theorem (Herman, Putnam, Skau (1992))
Each Cantor minimal system (X , T ) is conjugate to a
Bratteli-Vershik system (XB , ϕB ).
Theorem (Medynets (2006))
Each Cantor aperiodic system (X , T ) is conjugate to a
Bratteli-Vershik system (XB , ϕB ).
We cannot control the cardinality of the set of minimal paths
in B.
Bratteli diagrams for Cantor aperiodic systems
Definition of a substitution
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Fix a finite alphabet A. Denote by A+ the set of all finite
words over A.
Definition
A substitution is a map σ : A → A+ .
Example
A substitution of a Chacon type
A = {0, s, 1}

 0 7→ 00s0
s 7→ s
σ:

1 7→ 0110
Bratteli diagrams for Cantor aperiodic systems
Definition of a substitutional dynamical system
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Given a substitution σ : A → A+ .
Denote the language of the substitution by L(σ), i.e. the set
of all words w ∈ A+ such that w ≺ σ n (a) for some a ∈ A
and n > 0.
Set Xσ = {x ∈ AZ : x[−n, n] ∈ L(σ)}.
Let Tσ denote the shift over AZ
Definition
(Xσ , Tσ ) is a substitutional dynamical system.
Bratteli diagrams for Cantor aperiodic systems
Example
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures

 0 7→ 00s0
s 7→ s
σ:

1 7→ 0110
Bratteli diagrams for Cantor aperiodic systems
Example
Cantor
Aperiodic
Systems
v0
Bratteli
diagrams
Invariant
measures

 0 7→ 00s0
s 7→ s
σ:

1 7→ 0110
w1 w2 w3 w4 w5 w6 w7 w8
Bratteli diagrams for Cantor aperiodic systems
Classes of Bratteli diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Theorem (Durand, Host, Skau)
The Bratteli-Vershik system (XB , ϕB ) is stationary and
minimal if and only if ϕB is either an odometer or a primitive
substitution system.
Theorem (Bezuglyi, Kwiatkowski, Medynets)
The Bratteli-Vershik system (XB , ϕB ) is stationary and
aperiodic if and only if ϕB is either an “almost” odometer or
an aperiodic substitution system.
Invariant measures
Existence
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Let T : X → X be a homeomorphism of a Cantor set X .
Denote by MT (X ) the set of Borel probability T -invariant
measures on X , i.e.,
µ ∈ MT (X ) iff µ(B) = µ(T −1 B) for any Borel B.
Invariant measures
Existence
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Let T : X → X be a homeomorphism of a Cantor set X .
Denote by MT (X ) the set of Borel probability T -invariant
measures on X , i.e.,
µ ∈ MT (X ) iff µ(B) = µ(T −1 B) for any Borel B.
Theorem
MT (X ) 6= ∅.
Invariant measures
Invariant for orbit equivalence
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
The simplex MT (X ) is an invariant for conjugacy and
orbit equivalence.
Invariant measures
Invariant for orbit equivalence
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
The simplex MT (X ) is an invariant for conjugacy and
orbit equivalence.
The set
{µ(E) : E is clopen in X and µ ∈ MT (X )}
is an invariant for conjugacy and orbit equivalence.
Invariant measures
Invariant for orbit equivalence
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
The simplex MT (X ) is an invariant for conjugacy and
orbit equivalence.
The set
{µ(E) : E is clopen in X and µ ∈ MT (X )}
is an invariant for conjugacy and orbit equivalence.
Theorem (Giordano, Putnam, Skau)
If (X , T ) is minimal and uniquely ergodic, then the set
{µ(E) : E is clopen in X } is a complete invariant of orbit
equivalence. Here µ is the unique invariant measure.
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
µ is an invariant
measure.
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
µ is an invariant
measure.
Invariant
measures
v ∈ Vn and e ∈ E(v0 , v ).
v
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
µ is an invariant
measure.
Invariant
measures
v ∈ Vn and e ∈ E(v0 , v ).
(n)
v
Denote by Xv (e) the
set {x : x[1, n] = e}.
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
µ is an invariant
measure.
Invariant
measures
v ∈ Vn and e ∈ E(v0 , v ).
(n)
v
Denote by Xv (e) the
set {x : x[1, n] = e}.
(n)
pv
(n)
= µ(Xv (e)).
(n)
pn = (pv )v ∈Vn .
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
(n)
pv
Invariant
measures
(n)
= µ(Xv (e)).
(n)
pn = (pv )v ∈Vn .
v
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
(n)
pv
Invariant
measures
(n)
= µ(Xv (e)).
(n)
pn = (pv )v ∈Vn .
(n)
v
Then pv =
(n+1)
(n+1)
(n+1)
p1
+ p2
+ 2p3
.
Invariant measures
Stationary diagrams
Cantor
Aperiodic
Systems
Bratteli
diagrams
(n)
pv
Invariant
measures
(n)
= µ(Xv (e)).
(n)
pn = (pv )v ∈Vn .
(n)
v
Then pv =
(n+1)
(n+1)
(n+1)
p1
+ p2
+ 2p3
.
pn = Apn+1 , here A is
the incidence matrix
between levels n and
n + 1.
Invariant measures
Main conditions
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
1
pn = Apn+1
Invariant measures
Main conditions
Cantor
Aperiodic
Systems
Bratteli
diagrams
1
pn = Apn+1
Invariant
measures
2
pn ∈ RN
+ , here N is the dimension of the matrix.
Invariant measures
Main conditions
Cantor
Aperiodic
Systems
Bratteli
diagrams
1
pn = Apn+1
Invariant
measures
2
pn ∈ RN
+ , here N is the dimension of the matrix.
P
(1)
v ∈V1 pv = 1.
3
Invariant measures
Main conditions
Cantor
Aperiodic
Systems
Bratteli
diagrams
1
pn = Apn+1
Invariant
measures
2
pn ∈ RN
+ , here N is the dimension of the matrix.
P
(1)
v ∈V1 pv = 1.
3
Theorem
There is one-to-one correspondence between invariant
measures and sequences of vectors satisfying Conditions
(1), (2), and (3).
Invariant measures
Core of the matrix: Perron-Frobenius Theory
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
pn = Apn+1 and pn ∈ RN
+ imply
T
pn ∈ k ≥1 Ak RN
+ for every k ≥ 1.
Invariant measures
Core of the matrix: Perron-Frobenius Theory
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
pn = Apn+1 and pn ∈ RN
+ imply
T
pn ∈ k ≥1 Ak RN
+ for every k ≥ 1.
T
Set core(A) = k ≥1 Ak RN
+.
Invariant measures
Core of the matrix: Perron-Frobenius Theory
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
pn = Apn+1 and pn ∈ RN
+ imply
T
pn ∈ k ≥1 Ak RN
+ for every k ≥ 1.
T
Set core(A) = k ≥1 Ak RN
+.
Theorem (Geometrical P.-F. Theory)
A(core(A)) = core(A). Furthermore, A|span(core(A)) is
invertible.
Invariant measures
Core of the matrix: Perron-Frobenius Theory
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
pn = Apn+1 and pn ∈ RN
+ imply
T
pn ∈ k ≥1 Ak RN
+ for every k ≥ 1.
T
Set core(A) = k ≥1 Ak RN
+.
Theorem (Geometrical P.-F. Theory)
A(core(A)) = core(A). Furthermore, A|span(core(A)) is
invertible.
Corollary
(1) There is one-to-one correspondence between invariant
measures and vectors
p1 from core(A) ∩ ∆, where
P
:
x
=
1}.
∆ = {x ∈ RN
i
+
(2) Ergodic measures correspond to extreme vectors.
Invariant measures
Description of the cone
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Theorem (Geometrical P.-F. Theory)
(1) The cone core(A) is simplicial and polyhedral (finitely
generated).
(2) Extreme vectors of core(A) are eigenvectors of Ai ,
i = 1, . . . , N.
Corollary
Each system whose Bratteli diagram is stationary has only a
finite number of ergodic measures.
Invariant measures
Example: one ergodic measure
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures

1 1 0
A= 1 1 2 
0 0 2

Invariant measures
Example: one ergodic measure
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures

1 1 0
A= 1 1 2 
0 0 2

x = (1/2, 1/2, 0)
and
λ=2
core(A) = {βx : β ≥ 0}.
Invariant measures
Example: one ergodic measure
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
1
2
1
2
0

1 1 0
A= 1 1 2 
0 0 2

x = (1/2, 1/2, 0)
and
λ=2
core(A) = {βx : β ≥ 0}.
Invariant measures
Example: one ergodic measure
Cantor
Aperiodic
Systems
Bratteli
diagrams

1 1 0
A= 1 1 2 
0 0 2

Invariant
measures
1
2λ
1
2λ
0
x = (1/2, 1/2, 0)
and
λ=2
core(A) = {βx : β ≥ 0}.
Invariant measures
Example: two ergodic measures
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures
Invariant measures
Example: two ergodic measures
Cantor
Aperiodic
Systems
Bratteli
diagrams
Invariant
measures

1 1 0
A= 1 1 2 
0 0 3

x1 = (1/2, 1/2, 0) and λ1 = 2
x2 = (2/9, 4/9, 3/9) and
λ2 = 3.