Second-Order Ergodic Theorem for Substitution
Tiling Systems
Boris Solomyak and Konstantin Medynets
United States Naval Academy
Annapolis, Maryland
Ergodic Theorems
Let T be an ergodic measure-preserving transformation on a finite
measure space (X , µ).
Theorem (Birkhoff, Hopf)
For any f ∈ L1 (X , µ),
f (x) + f (Tx) + · · · + f (T n−1 x)
→
n
Z
fd µ.
X
Suppose that the measure
µ is infinite, σ-finite . Then for any
R
1
f , g ∈ L (X , µ) with X gdµ 6= 0, we have that
R
fd µ
f (x) + f (Tx) + · · · + f (T n−1 x)
→ RX
.
n−1
g (x) + g (Tx) + · · · + g (T
x)
X gdµ
Renormalization Constants
Hopf’s Ergodic Theorem:
f (x) + f (Tx) + · · · + f (T n−1 x)
→
Sn g (x)
Z
fd µ,
X
where Sn g (x) = g (x) + g (Tx) + · · · + g (T n−1 x).
Remark
There are no constants an > 0 such that
f (x) + f (Tx) + · · · + f (T n−1 x)
→
an
for all f ∈ L1 (X , µ).
Z
X
fd µ
Albert Fisher’s Integer Cantor Sets.
Denote by X ⊂ {0, 1}Z the orbit-closure, under the left-shift T , of
of the sequence
ω = 101000101000000000101 . . .
The sequence is generated by the substitution σ(1) = 101 and
σ(0) = 000.
The dynamical system (X , T ) has a unique minimal component,
which is also a fixed point, M = {. . . 00.00 · · · }.
Integer Cantor Set. Bratteli-Vershik Model.
The system also admits a
unique ergodic probability
measure, which is
automatically supported by the
minimal component.
There is a unique (up to
scaling) σ-finite invariant
measure µ on X such that
0 < µ(U) < ∞ for at least one
open set U.
Albert Fisher’s Second Order Ergodic Theorem.
Theorem (Fisher, 1993)
Let (X , T ) be the Integer Cantor Dynamical System and µ be the
invariant measure normalized so that µ([1]) = 1. Then for every
f ∈ L1 (X , µ) and µ-a.e. x ∈ X ,
n
1 X f (x) + f (Tx) + · · · + f (T k−1 x)
=
n→∞ log(n)
ck α+1
lim
k=1
Z
fd µ.
X
Here
α = log(2)/ log(3) is the Hausdorff dimension of the Cantor
set;
c is the logarithmic density of the Cantor set.
Plan of the proof.
Prove the second-order
ergodic theorem for the
indicator function of a
clopen set. Hopf’s ratio
ergodic theorem will allow
us to deduce the result for
any function.
Associate a finite
measure-preserving
Markov shift. The shift
will expand a “window”
about the zero coordinate
for any tiling. This will
allow us to count # of
1’s.
Infinite Measure Preserving Substitution Systems
Consider a substitution system σ : F → F. Consider the
substitution matrix M(σ). Example. σ(0) = 000 and σ(1) = 101.
Then
M(σ) =
3 1
0 2
To get a “nice”1 infinite measure, the substitution matrix has to
be upper-triangular
M(σ) =
A C
0 B
with ρ(A) > ρ(B).
1
A measure that is finite on a clopen set.
Infinite Measure Preserving Substitution Systems
Let ξ = (ξa ) be the left Perron-Frobenius eigenvector for
M(σ). Note that ξa > 0.
S
For each x ∈ X , define a tiling I (x) = n∈Z Ixn , where Ia is a
translation of [0, ξa ]. For σ(0) = 000 and σ(1) = 101, each
interval has length 1.
Consider the flow on tilings X̄ = {I (x) + t : x ∈ X , t ∈ R}.
This is a substitution tiling system or flow under the
function. The results work for practically any finite-local
complexity substitution tiling system.
Fractals associated to substitution systems
Note that ρ(A)Ia = ⊔v ∈σ(a) Iv . For example,
[0, 3]1 = [0, 1]1 ∪ [1, 2]0 ∪ [2, 3]1 .
So, Ia = ρ(A)−1 (⊔v ∈σ(a) Iv ). For example,
[0, 1]1 = [0, 1/3]1 ∪ [1/3, 2/3]0 ∪ [2/3, 1]1 .
Remove the intervals corresponding to the matrix A in
A C
M(σ) =
.
0 B
For example, [0, 1]1 7→ [0, 1/3]1 ∪ [2/3, 1]1 .
Repeat the process for each rescaled subinterval. This will
converge to a fractal Ca , so called graph-iterated function
systems (Mauldin-Williams, 1988).
Markov Chain Associated to Fractals
.
S
For each x ∈ X , consider the fractal tiling C(I (x)) = n∈X Cxn
with blocks corresponding to letters in A missing.
It turns out that σ : {C(I (x)) + t : x ∈ X , t ∈ Cb } is a
Markov chain.
Let α be the Hausdorff dimension of the fractals {Cb }, b ∈ B.
Then the Hausdorff measure Hα (Cb ) > 0 if and only if b ∈ B.
To count the number of 1’s, it is enough to estimate
Hα (BR ∩ C(I (x))), where BR is the ball of radius R.
The second-order averages become
Z z α
k−1
1X
H (BR ∩ C(I (x)))
1
F (σ −1 C(I (x))),
dR
=
lim
lim
z→∞ log(z) 1
k→∞ k
R α+1
i =0
for some F , an integrable function on the space of fractal
tilings.
Main Theorem
Let (Ω, µ, Rd ) be a tiling dynamical system corresponding to a tile
substitution σ and an expansion map ϕ = λ · O. Here O is an
orthogonal matrix. Then for every f ∈ L1 (Ω, µ), µ-a.e. tiling T ,
Z
Z t R f (T − u)du
dR
1
BR
fd µ,
lim
=
t→∞ log(t) 1
c(2R)α
R
Ω
Here
α = log(ρ(B))/ log(λ). Note that in 1-dimensional case,
λ = ρ(A).
c is the logarithmic density of fractals
1
k→∞ k
lim
Z
k
0
Hα (Bλ−t (u) ∩ Cb )
dt.
(2λ−t )α
Applications
The Hausdorff dimension α and logarithmic density c are
invariants of dynamical systems.
(α-dimensional frequency) Let (Xσ , µ, T ) be an infinite
measure-preserving dynamical system. Then for every letter
b ∈ B, the limit
1
n→∞ log(n)
lim
X
1≤k≤n,xk =b
exists is positive for µ-a.e. x ∈ Xσ .
1
kα