Dynamics on Bratteli Diagrams
Kostya Medynets
Wright State University
April 22, 2016
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Autonomous Differential Equations
• The Lennard-Jones Oscillator
• Consider an autonomous differential
equation
x′ = f (x),
x′ = p,
x ∈ X ⊂ Rn
p′ = 12(x−13 − x−7 )
(1)
• Fix a point x0 ∈ X . Denote by
φ(t, x0 ) the value x(t), where x(t)
is the unique solution of (1) subject
to x(x) = x0 .
• Group Property:
φ(t2 , φ(t1 , x0 )) = φ(t1 + t2 , x0 ).
• Flow on the phase space
Φt (x) = φ(x, t),
Φt : X → X
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Discrete Dynamical Systems
• Let Φt : X → X be a flow on the
phase space. For example, the
phase space could consist of all
possible positions and velocities of
a particle.
• Set T = Φ1 : X → X .
• Then T n (x0 ) represents the
position/velocity of the particle
whose initial location/velocity was
x0 n time units into the future.
• The transformation T : X → X is
the called the evolution law.
• Abstract Dynamical Systems: Let
X be a compact metric space and
T : X → X be a homeomorphism.
The pair (X, T ) is called a
•
•
(discrete) dynamical system.
(X, T ) is called minimal if every
T -orbit, {T n (x0 ) : n ∈ Z} is
dense in X .
(Zorn’s Lemma:) Every system has
a minimal subsystem.
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Example: Irrational Rotation
• Let X = S1 and T : X → X be the rotation of X by an angle θ.
• If θ/(2π) ∈ Q, the T -orbits are finite.
• If If θ/(2π) ∈ R \ Q, each T -orbit is infinite and dense. The system is
minimal.
Theorem 1. If a dynamical system (X, T ) is equicontinuous,
dist(x, y) < δ implies dist(n (x), T n (x)) < ε, then (X, T ) is conjugate to
a translation on a compact Abelian group [Auslander, Ellis].
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Symbolic Systems
• Phase Space. Let A = {0, 1, . . . , n − 1}. Consider Xn the space of
two-sided sequences with values in A.
• A is given the discrete topology and Xn the product topology. Two
sequences are close if the share a long common word centered at 0. Xn
is the Cantor set.
• Dynamics. Let T : Xn → Xn be the left Bernoulli shift. For example, for
n = 2,
T (· · · 101|001 · · · ) = · · · 1010|01 · · · .
• Subshifts. Let Y ⊂ Xn be closed and shift-invariant, that is,
T (Y ) = Y . Then (Y, T ) is called a subshift.
Theorem 2. A dynamical system (X, T ) is topologically conjugate to a
subshift if it is expansive, that is, there exists c > 0 such that for any
x 6= y ∈ X , dist(T n (x), T n (y)) > c for some n ∈ Z [Perry].
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Lind-Marcus: Applications to Coding
•
•
•
•
Computers store data as 0’s and 1’s.
Polarity reversals along the track generate voltage pulses.
A pulse is a 1. The absence of a pulse is a 0.
Problems: Intersymbol Interference and Clock Drift.
• Fix: no more than consecutive n 1’s and m 0’s. Gives rise to a subshift.
• A subshift is good for coding/decoding if its entropy ≥ log(2).
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Bratteli diagrams: Definition
• Bratteli diagram = infinite graded
•
•
•
•
•
graph
Single root
Edges connect consecutive
levels
No isolated vertices
Phase Space = path space of
infinite paths
Phase Space is a Cantor set
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Bratteli Diagrams: Ordering
•
•
•
•
3
0
Consider an arbitrary vertex.
Consider all incoming edges.
Enumerate these edges.
Repeat for every vertex.
21
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Bratteli Diagrams: Max Paths
• x = (xn ) is maximal if every
•
0
0
1
1
1
1
0
3
0
3
0
12
0
12
•
edge xn is maximal amongst the
edges with the same range.
Denote by Xmax and Xmin the
set of all maximal and minimal
paths.
Xmax 6= ∅ and Xmin 6= ∅.
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Bratteli Diagrams: Action
• Define T : X → X.
• Fix x ∈ X .
• Find the first non-maximal edge
xk .
0
1
1
0
3
0
12
• Take the successor of xk .
• Connect the successor to the
top vertex by the unique path
labeled with 0’s.
• T : X \ Xmax → X \ Xmin is
0
1
1
0
3
0
12
continuous.
If T can be extended to T : X → X , (X, T ) is called a Bratteli-Vershik
system.
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Bratteli-Vershik Models
Theorem 3 (Herman-Putnam-Skau 1995). Every minimal system (X, T ) on
the Cantor set X is conjugate to a minimal Bratteli-Vershik system.
Minimal systems correspond to simple Bratteli diagrams.
Theorem 4 (M. 2005). Every aperiodic, no finite orbits, (X, T ) is conjugate
to a Bratteli-Vershik system.
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Examples: Odometer
• Let Z2 be the group of 2-adic
•
•
integers, X2 ∼
= {0, 1}Z .
T : Z2 → X2 , T (x) = x + 1.
Consider x0 = (0, 0, 0, . . .)
T (x0 ) = (1, 0, 0, . . .)
T 2 (x0 ) = (0, 1, 0, . . .)
• (Z2 , T ) is called an odometer.
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Examples: Stationary Diagrams
Theorem.
Dynamics on stationary Bratteli diagrams correspond to either odometers or
to substitution dynamical systems [Durand, Host, Skau, 1999, minimal
systems] and [Bezuglyi, Kwiatkowski, M., 2009, general aperiodic systems].
• Let A be a finite alphabet and σ : A → A+ . Let Xσ be the set of
sequences x ∈ AZ such that every x[−n, n] is a subword of σ k (a) for
some a ∈ A, k > 0. Then (Xσ , T ) is a substitution system.
• Example (Fibonacci Substitution) σ(0) = 01, σ(1) = 0.
0 7→ 01 7→ 010 7→ 01001 · · · 7→ Fibonacci Word.
• The length of σ n (0) = Fn+2 , the (n + 2)th Fibonacci number.
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Examples: Stationary Diagrams (Fibonacci)
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Invariant Measures
• Consider an autonomous Hamiltonian system
x′ = f (x),
x ∈ X ⊂ Rn
and the associated one-parameter flow
Φt : X → X.
• Lioville’s Theorem asserts if D ⊂ X , open, then
V ol(D) = V ol(Φt (D)) for every t.
• Let (X, T ) be a dynamical system.
• A probability measure µ on X is called T -invariant if µ(A) = µ(T (A))
for every Borel set A ⊂ X .
• The invariant measure µ is ergodic if µ(A) = 0, 1 for any Borel set A
such that T (A) = A.
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Ergodic Theorem
• Invariant measures form a simplex.
• Ergodic measures are precisely extreme elements of that simplex.
• Any measure is a convex combination (possibly, infinite uncountable) of
ergodic measures.
Theorem 5 (Birkhoff). Let (X, T ) be a dynamical system and µ be an
ergodic measure. Then for every f ∈ L1 (X, µ) and almost every x ∈ X ,
Z
n−1
X
1
lim
f (T k x) =
f (x)dµ(x).
n→∞ n
X
k=0
In particular, if f = 1A , then
card{0 ≤ k ≤ n − 1 : T k (x) ∈ A}
= µ(A).
f req(x, A) = lim
n→∞
n
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Invariant Measures on Bratteli Diagrams
• µ is an invariant measure.
• v ∈ Vn and e ∈ E(v0 , v).
(n)
• Denote by Xv (e) the set
{x : x[1, n] = e}.
(n)
(n)
• pv = µ(Xv (e)).
(n)
pn = (pv )v∈Vn .
v
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Invariant Measures on Bratteli Diagrams
(n)
• pv
(n)
= µ(Xv (e)).
(n)
pn = (pv )v∈Vn .
• Then
(n)
(n+1)
(n+1)
(n+1)
+p2
+2p3
.
pv = p1
• pn = An pn+1 , here An is the
v
incidence matrix between levels
n and n + 1.
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Invariant Measures: Perron-Frobenius Theory
Assume that {An } are N × N , a uniformly bounded diagram.
• pn = An pn+1
• pn ≥ 0 .
Set core(An ) =
T
N
A
A
·
·
·
A
R
n
n+1
n+k
+.
k≥1
Theorem 6 (Geometrical P.-F. Theory). (1) The cones core(An ) are
polyhedral (finitely generated).
(2) The number of extreme rays does not exceed N .
(3) Ergodic measures correspond to sequences of extreme vectors {p̄n }.
Corollary 7. For stationary diagrams, when A = An , ergodic measures
correspond to non-negative eigenvectors of A.
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Invariant Measures: Example

1
2
1
2

1 1 0
A= 1 1 2 
0 0 2
0
x = (1/2, 1/2, 0)
1
2λ
1
2λ
0
and
λ=2
core(A) = {βx : β ≥ 0}.
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Invariant Measures: Example 2
Consider two Bratteli diagrams determined by incidence matrices
An =
2
n
1
1
n2
and Bn =
n 1
1 n
.
{An } has two ergodic measures, whereas {Bn }n≥1 is uniquely ergodic.
Matrices are contractions on the positive projective space (Birkhoff’s
contraction coefficient).
• Handelman (1999), Markov chains (uniqueness of stationary
distributions).
• Bezuglyi, Kwiatkowski, Medynets, Solomyak (2014), Trans. AMS.
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Bratteli diagrams: Mixing Properties
• If a Bratteli diagram has K vertices at every level, then the associated
system (for any ordering) has fewer than K + 1 ergodic measures.
• A measure-preserving dynamical system (X, T, µ) is called strongly
mixing if for any two sets A, B ⊂ X , µ(A ∩ T n (B)) = µ(A)µ(B) as
n → ∞. In other words, any two sets become asymptotically
independent. Strong mixing implies ergodicity.
• A full Bernoulli shift is strongly mixing. One can derive the Strong Law of
Large Numbers from the (pointwise) Ergodic Theorem for Bernoulli shifts.
Theorem 8 (Bezuglyi-Kwiatkowski-M.-Solomyak, 2014). Dynamical systems
on uniformly bounded (finite rank) Bratteli diagrams cannot be strongly
mixing.
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