Lecture 30:
Recall: defns of Φ, f Φ, ZnΦ.
Recall: Proposition: (a special case of Ruelle, Theorem 3.4) For
n.n. Zd SFT M and a n.n. potential Φ,
P (Φ) = P (σ|M , f Φ) = lim (1/nd) log ZnΦ
n→∞
Recall: Perron-Frobenius Theorem:
Let A be an irreducible matrix with spectral radius
λA := max eigenvalues λ of A |λ|. Then
1. λA > 0 and is an eigenvalue of A.
2. λA has a (strictly) positive eigenvector and is the only eigenvalue
with a nonnegative eigenvector.
Proposition: For a TMC MC (i.e., n.n. SFT with d = 1 defined
by 0-1 matrix C) with C irreducible and a n.n. potential Φ (defined
on the non-forbidden configurations on edges), define the matrix C Φ
by:
(C Φ)ij = Cij e−Φ(ij)
Then
P (Φ) = P (σ|MC , f Φ) = log λC Φ
Corollary: h(σ|MC ) = log λC .
Proof:
ZnΦ = 1(C Φ)n−11
So,
P (Φ) = lim (1/n) log ZnΦ = lim (1/n)1(C Φ)n1
n→∞
n→∞
P-F Theorem guarantees a positive (right) eigenvector v corresponding to λ = λC Φ .
1
Claim: There exist constants K1, K2 > 0 s.t.
K1λn ≤ 1(C Φ)n1 ≤ K2λn
Proof of Claim:
1(C Φ)n1 ≤ 1(C Φ)n(
1(C Φ)n1 ≥ 1(C Φ)n(
v
1
) = (1 · v)
λn
min(v)
min(v)
1
v
) = (1 · v)
λn
max(v)
max(v)
Take log, divide by n, take limn, to get:
P (Φ) = P (σ|MC , fΦ) = log λC Φ
Example: Golden Mean SFT, Φ = 0.
C=
1 1
1 0
√
Eigenvalues are log(1 ± 5)/2.
√
h(σ|MC ) = P (σ, 0) = log λC = log(1 + 5)/2.
Example: 1D Ising (ferromagnetic, no external field, coupling
strength = 1, inverse temperature = β)
Here Φ(00) = −β = Φ(11) and Φ(01) = β = Φ(10) and
1 1
C=
1 1
and so
β −β e e
CΦ =
e−β eβ
Eigenvalues are eβ ± e−β .
2
P (Φ) = log λC Φ = log(eβ + e−β ).
Theorem (Variational Principle for irreducible TMC) Let C be
irreducible, Φ be nn potential, and M = M(σ|MC ). Then
Z
P (Φ) = log λC Φ =
sup
hµ(σMC ) + f Φdµ
µ∈M(σ|M )
C
and the sup is achieved uniquely by the stationary Markov chain:
vj
Pij = CijΦ
λC Φ vi
where C Φv = λC Φ v.
Special case: Φ = 0: topological entropy is the sup of measuretheoretic entropies.
Idea of Theorem: the measure µΦ that achieves the sup is a staΦ
tionary measure that satisfies µΦ(w) ∼ e−U (w) for configurations
w on finite subsets of Zd (by µΦ(w), we mean the measure of the
cylinder set corresponding to w). So, if Φ = 0, µΦ should be the
“most uniform” stationary measure with support contained in MC .
Example: Golden Mean, Φ = 0.
v = [λ, 1]T . So,
P =
1/λ 1/λ2
1
0
where λ is the golden mean.
Example: Ising Model:
v = [1, 1]T . So,
P =
1
eβ + e−β
3
β
−β
e e
e−β eβ
Lecture 31:
Recall:
Theorem (Variational Principle for irreducible TMC) Let C be
irreducible, Φ be nn potential, and M = M(σ|MC ). Then
Z
hµ(σMC ) + f Φdµ
P (Φ) = log λC Φ =
sup
µ∈M(σ|M )
C
and the sup is achieved uniquely by the stationary Markov chain:
vj
Pij = CijΦ
λC Φ vi
where C Φv = λC Φ v.
Special case: Φ = 0: topological entropy is the sup of measuretheoretic entropies.
Proof of Variational Principle:
Step 1: For all µ ∈ M(σ|MC ),
Z
hµ(σMC ) + f Φdµ ≤ log λC Φ = P (σ|MC , f Φ).
Lemma:
−
max P
(p1 ,...,pn ):pi ≥0, i pi =1
X
pi log pi +
i
X
X
piai = log(
e ai )
i
with equality iff pi = eai /K, where K =
i
ai
ie .
P
Proof of Lemma: Let
eai
pi K
αi = , xi = a
K
ei
Then
X
αi = 1,
X
i
i
4
αixi = 1.
By Jensen applied to φ(x) = x log x,
X
X
pi K
0 = φ(1) ≤
αiφ(xi) =
pi log a
ei
i
i
X
X
piai + log K
pi log pi −
=
i
i
Proof of Step 1: Let α = {Ai} be the time-0 partition: Ai = {x ∈
MC : x0 = i}. Then
Z
X
n−1
Φ
Hµ(α0 ) + Snf dµ ≤
−µ(A) log µ(A) + µ(A)Snf Φ(xA)
A∈α0n−1
(where xA achieves the max of Snf Φ on A). By the Lemma, this is
X
Φ
≤ log(
eSnf (xA))
A∈α0n−1
So
Z
hµ(σMC ) +
f Φdµ = lim (1/n)(Hµ(α0n−1) +
Z
n→∞
≤ lim (1/n) log(
n→∞
X
eSnf
Φ (x
A)
Snf Φdµ)
)
A∈α0n−1
= P1(σ|MC , f Φ) = P (σ|MC , f Φ)
Step 2:Let µ ∈ M(σ|M ). Let S = {i : µ(Ai) > 0} (recall
Ai = {x ∈ MC : x0 = i}). Let
Aij = {x ∈ MC : x0 = i, x1 = j}
and
Pij =
µ(Aij )
, i, j ∈ S,
µ(Ai)
5
πi = µ(Ai), i ∈ S
(here, P is a transition probability matrix, not pressure). Then P
defines a first-order stationary Markov chain ν with stationary vector
π and h(µ) ≤ h(ν) with equality iff µ = ν.
ν is called the Markovization of µ.
Proof: Check πP = π
−i
−1
hµ(σC ) = Hµ(α | ∨∞
i=1 σ (α)) ≤ Hµ (α|σ (α))
= Hµ(σ −1(α) ∨ α) − Hµ(σ −1(α)) = Hµ(σ −1(α) ∨ α) − Hµ(α)
X
µ(Aij )
−1
µ(Aij ) log(
= Hµ(σ (α)|α) = −
)
µ(A
)
i
ij
X
=−
πiPij log Pij = hν (σC )
ij
−i
By entropy inequality 10, equality holds iff α ⊥σ−1(α) ∨∞
i=2 σ (α).
i.e., µ is Markov, equivalently µ = ν.
Step 3: Unique measure that achieves the sup.
From Steps 1 and 2, we know that the sup is ≤ log λ and the
only invariant measures that can achieve the sup must be first-order
stationary Markov.
Step 3: The sup is log λ, it is achieved by the the first order
stationary Markov chain µ defined by
vj
Pij := CijΦ ,
λvi
and it is achieved uniquely.
Proof: Recall that v is a right eigenvector for λ = λC Φ . Let w be
a strictly positive left eigenvector for λ, normalized s.t. w · v = 1.
Then π, defined by πi = wivi , is the stationary vector for P :
X
X
vj
(πP )i =
πiPij =
wiviCijΦ
λvi
j
j
6
= wi
X
CijΦ
j
vj
= w i vi = π i
λ
Note that
− log Pij − Φij = − log CijΦ
vj
− Φij = log λ + log vi − log vj
λvi
provided Cij > 0. Thus, for any invariant measure ν on MC ,
Eν (− log Pij −Φij ) = log λ+Eν (log vi−log vj ) = log λ+Eν (g◦σ−g) = log λ
where g(x) = − log vx0
(Note that since ν is a measure on MC , whenever ν(Aij ) > 0, we
have Cij > 0).
In particular
Z
X
X
Φ
hµ(σ|M )+ f dµ = −
πiPij log Pij −
Φ(ij)πiPij = Eµ(− log Pij −Φi
ij
ij
and so µ achieves the sup.
If ν is another first order stationary Markov chain on MC , defined
by P 0, π 0, which achieves the sup, then
Z
Eν (− log Pij0 − Φij ) = hν (σ|M ) + f Φdν = log λ
So,
Eν (− log Pij0 − Φij ) = log λ = Eν (− log Pij − Φij )
So,
Eν (
log Pij
) = 0.
log Pij0
But
X
X
P
0 0
0
log Eν 0 = log
πiPij Pij /Pij = log
πi0 Pij = 0.
P
ij
ij
7
Thus,
P
P
=
0
=
log
E
ν
P0
P0
By Jensen, equality happens only if P 0 = P for all i, j s.t. Pij0 > 0.
But Pij0 > 0 iff Pij > 0:
if Pij0 > 0, then Pij > 0 since ν is a measure on MC ;
P
P
1 = j:P 0 >0 Pij0 = j:P 0 >0 Pij and so if Pij0 = 0, then Pij = 0.
ij
ij
0
So, P = P and ν = µ. Eν log
8
Lecture 32:
General Variational Principle (Saff): Let T be a continuous Zd+
action on a compact metric space M and f : M → R be a continuous
function. Then
Z
P (T , f ) = sup hµ(T ) + f dµ
µ∈M(T )
Last time: special case where the supremum is achieved uniquely
by an explicit, simple µ (1st-order stationary Markov chain).
Example: There exists T , f s.t. the sup in the variational principle
is not achieved.
T will be a Z-action, i.e. continuous map compact metric M and
f = 0. So, P (T, f ) = h(T ).
For i = 1, 2, . . ., let Ti be a sequence of continuous maps on compact metric spaces Mi, with corresponding metrics ρi, s.t. h(Ti) is
strictly increasing to a limit L < ∞.
We can assume that the ρi-diameter of each Mi is 1 because the
diameter of each Mi is finite and so we can rescale each metric to
have diameter 1. Continuity of each Ti is not affected by rescaling.
Let M be the disjoint union of the Mi and one more point M0 =
{?}. Define T : M → M by T |Mi = Ti and T (?) = ?. Endow M
with metric ρ:
For i 6= 0 and x, y ∈ Mi, ρ(x, y) := (1/i2)ρi(x, y).
P
For 0 < i < j, x ∈ Mi, y ∈ Mj , ρ(x, y) := jk=i(1/k 2).
P
2
For i 6= 0, x ∈ Mi, d(x, ?) = ∞
k=i (1/k ).
Fact: T : M → M is continuous (because Mi → ?, T (Mi) = Mi)
and T (?) = ?.
9
Proposition: Let T : M → M be a continuous map and for
i = 1, 2, . . ., µi ∈ M(T ). Let p be a countably infinite probability
P
vector: p = (p1, p2, . . .), with each pi ≥ 0 and i pi = 1. Then
X
P
pihµi (T ).
h i piµi (T ) =
i
Proof of Proposition: more general version (replace {µi} with a
continuum of measures) in Theorem 4.3.7 of Keller; more special
version (replace {µi} with just two measures) in Theorem 8.1 of
Walters. See below. Proof that the sup in the variational principal is not achieved:
Let µi := (µ|Mi )/(µ(Mi)) if µ(Mi) 6= 0. Then
µ=
X
pi µi
{i:µ(Mi )6=0}
where pi = µ(Mi). So,
X
hµ(T ) =
pihµi (T ) ≤
X
pih(Ti) < L
{i:µ(Mi )6=0}
{i:µ(Mi )6=0}
Since Ti = T |Mi, each h(T ) ≥ h(Ti) for all i. Thus, h(T ) ≥ L.
Thus,
hµ(T ) < h(T ).
(in fact, one can show that h(T ) = L, but we don’t need that here).
Idea of preceding: There are invariant measures µi on each Mi
s.t. hµi (Ti) → L, but the sequence of supports of these measures
converges to a single point, which cannot carry any entropy.
Note: In fact, the argument above can be modified to only use
the Walters-Theorem 8.1 version (i.e., an average of two measures)
of the Proposition above, which we now prove:
10
hpµ+(1−p)ν (T ) = phµ(T ) + (1 − p)hν (T ).
Proof:
Let α be a finite measble partition. Since φ(x) = −x log x is
concave-down (φ00 < 0),
X
−(pµ(Ai)+(1−p)ν(Ai)) log(pµ(Ai)+(1−p)ν(Ai))
Hpµ+(1−p)ν (α) =
i
≥ p(
X
−µ(Ai) log µ(Ai)) + (1 − p)(
X
−ν(Ai) log ν(Ai))
i
i
= pHµ(α) + (1 − p)Hν (α).
So,
Hpµ+(1−p)ν (α) ≥ pHµ(α) + (1 − p)Hν (α).
Now,
[(pµ(A)+(1−p)ν(A)) log(pµ(A)+(1−p)ν(A))]−[pµ(A) log µ(A))]
−[(1 − p)ν(A) log ν(A)]
= pµ(A)[log(pµ(A) + (1 − p)ν(A)) − log(pµ(A))]
+(1 − p)ν(A))[log(pµ(A) + (1 − p)ν(A)) − log(1 − p)ν(A))]
+pµ(A)[log(pµ(A))−log µ(A)]+(1−p)ν(A)[log((1−p)ν(A))−log(ν(A))]
≥ 0 + 0 + pµ(A) log p + (1 − p)ν(A) log(1 − p)
(since log is increasing) So,
Hpµ+(1−p)ν (α) − (pHµ(α) + (1 − p)Hν (α)) ≤ log 2.
So,
hpµ+(1−p)ν (T, α) = phµ(T, α) + (1 − p)hν (T, α)
So,
hpµ+(1−p)ν (T ) ≤ phµ(T ) + (1 − p)hν (T )
11
Let > 0. Choose α1, α2 s.t.
hµ(T, α1) > hµ(T ) − , hν (T, α1) > hν (T ) − ,
Then,
hpµ+(1−p)ν (T, α1 ∨ α2) = phµ(T, α1 ∨ α2) + (1 − p)hν (T, α1 ∨ α2)
≥ phµ(T, α1) + (1 − p)hν (T, α2) > phµ(T ) + (1 − p)hν (T ) − Argue differently if entropy is infinite. Theorem: If a G-action T (where G is Zd+ or Zd) is expansive and
f is continuous, then the sup in the variational principle is achieved.
Cor: For the shift action σ|M on a (two-sided) shift space M and
continuous f , E(σ|M , f ) 6= ∅.
Outline of proof of Theorem:
Let α = {A1, . . . , Ak } be a finite Borel partition of M .
Defns: α is δ-fine if for each i, diam(Ai) < δ
Defn: For a Borel probability measure µ, α is µ-good if for each i,
µ(∂Ai) = 0.
Step 1(Theorem 4.5.6, Keller): Let δ be an expansive constant for
T . For a δ-fine partition of M , we have ∨g∈GT −g (α) = B, t he Borel
σ-algebra.
Step 2(Theorem 4.2.4, Keller): Let µ ∈ M(T ). If α is µ-good,
then the function M(T ) 7→ R, given by µ 7→ hµ(T , α) is upper
semi-continuous at µ, i.e., whenever µn → µ weakly, then
lim sup hµn (T , α) ≤ hµ(T , α)
n→∞
12
R
R
Weak convergence means: for all continuous f , f dµn → f dµ;
equivalently for all Borel A with µ(∂A) = 0, µn(A) → µ(A).
Step 3: (Lemma 4.4.10, Keller): For a compact metric space M , a
Borel probability measure µ on M , and δ > 0, there exists a δ-fine,
µ-good partition of M .
Note: Step 3 involves topology and measure theory, but has nothing to do with the dynamics.
Proof of Theorem:
Let µn ∈ M(T ) s.t. hµn (T ) +
R
f dµn → P (T , f ).
Note that M(T ) is compact in the weak topology: the space X
of all Borel probability measures is a weakly compact metric space,
and M(T ) is a weakly closed subset of X.
So, there is a subsequence ni s.t. µni converges weakly to some
µ ∈ M(T ).
Let δ be an expansive constant for T . By Step 3, there is a δ-fine,
µ-good partition α of M .
By Step 1 and the Sinai generator theorem, for each ν ∈ M(T ),
hν (T , α) = hν (T ). In particular hµ(T , α) = hµ(T ) and each
hµni (T , α) = hµni (T )
Then by Step 2,
Z
P (T , f ) = lim sup(hµni (T )+
i→∞
Z
f dµni ) = lim sup(hµni (T , α)+
i→∞
Z
≤ hµ(T , α) +
Z
f dµ = hµ(T ) +
13
f dµ
f dµni )