Random homogeneous beta-expansions andself-similar measures
Kevin G. Hare
Joint work with Kathryn E. Hare, Sascha Troscheit

Numeration, May 22nd, 2018

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

1

Beta Expansions

2Measures and Local dimension

3

Random homogeneous beta-expansions

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
Let 0 ∈ S be a finite subset of R and β > 1. For x ∈ R, write
x=∞Xaii=1βiai ∈ S.

We will say that a1 a2 . . . is a beta-expansion of x with respect
to a digit set S.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
Let 0 ∈ S be a finite subset of R and β > 1. For x ∈ R, write
x=∞
X
ai
i=1

βi

ai ∈ S.

We will say that a1 a2 . . . is a beta-expansion of x with respect
to a digit set S.
Definition
We define the support of a beta-expansion as{x | x has at least one β-expansion}

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Example (Lebesgue measure)Let β = 3 and S = {0, 1, 2}.
We see that the beta-expansion of x is the base 3
representation of x.
The support of this beta-expansion is [0, 1].
The Hausdorff dimension of the support is 1.
All x ∈ [0, 1] have at least 1 expansion, and at most 2expansions.
Almost all x ∈ [0, 1] have a unique expansion.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Example (Cantor Set)
Let β = 3 and S = {0, 2}.
We see that the beta-expansion of x is the base 3expansion of x.
We see that for x in the support, x has a base 3 expansion
containing only 0 and 2.
The support of this beta-expansions is the classical middle
third Cantor set.
The Hausdorff dimension of the support is log(2)/ log(3).
All x in the support have a unique expansion.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Example (Bernoulli Convolution)√Let β = 1+2 5 , the golden ratio and S = {0, 1}.
The support of this beta-expansion is [0, β].
The Hausdorff dimension of the support is 1.
The point x = 0 and x = β have exactly one
beta-expansion.
The point x = 1 has countably many beta-expansions.
The point x = β 31−1 has uncountably manybeta-expansions.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Let G ≈ 1.618033, qf ≈ 1.754877, and qc ≈ 1.787231.
Theorem (Glendinning, Sidorov, 2001)
If q ∈ (G, qf ] then there are exactly 2 non-trival x in thesupport with unique expansions.
If q ∈ (qf , qc ) then there are countably many x in the
support with unique expansions.
If q ∈ [qc , 2) then there are uncountably many x in the
support with unique expansions.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

1

Beta Expansions

2Measures and Local dimension

3

Random homogeneous beta-expansions

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
We define a self-similar measure on the support ofβ-expansions asnoXµ(E) = P (a1 a2 . . . ) :ai β −i ∈ E
where E is any Borel set and P is the product measure on thedigit set S.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
We define a self-similar measure on the support ofβ-expansions asnoXµ(E) = P (a1 a2 . . . ) :ai β −i ∈ E
where E is any Borel set and P is the product measure on thedigit set S.
This is a non-atomic self-similar measure.
For β = 3 and S = {0, 1, 2}, This is Lebesgue measure on[0, 1].
For β = 3 and S = {0, 2}, This is the Hausdorff measureon the middle third Cantor set.
√For β = 1+2 5 and S = {0, 1} this is know as the Bernoulli
convolution with respect to the golden ratio.
K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
Given a self-similar measure µ, by the upper local dimensionof µ at x in the support, we mean the number
dimloc µ(x) = lim sup
r →0+

log µ([x − r , x + r ]).
log rReplacing the lim sup by lim inf gives the lower local
dimension, denoted dimloc µ(x). If the limit exists, we call thenumber the local dimension of µ at x and denote this bydimloc µ(x).

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

ExampleLet β = 3 and S = {0, 1, 2}.
Then for all x in the support,
lim+r →0log µβ ([x − 1/3n , x + 1/3n ])log µβ ([x − r , x + r ])= limn→∞log rlog 1/3nlog 2/3n= limn→∞ log 1/3n=1Hence all points have local dimension 1.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

ExampleLet β = 3 and S = {0, 2}. Then for all x in the support,lim+r →0

log µβ ([x − 1/3n , x + 1/3n ])log µβ ([x − r , x + r ])
= lim
n→∞
log r
log 1/3nlog 1/2n= limn→∞ log 1/3nlog 2=≈ 0.6309
log 3

Hence all points have local dimension log 2/ log 3.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

ExampleLet β ≈ 1.618 be the golden ratio and S = {0, 1}.
Feng - 2005, 2009: The set of attainable local dimensionsof µ is the interval, ≈ [0.9404, 1.4404].
Alexander, Zagier - 1991: Most points have localdimension ≈ 0.9957.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

1

Beta Expansions

2

Measures and Local dimension

3

Random homogeneous beta-expansions

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

ExampleLet β = 3. Let S1 = {0, 1, 2} and S2 = {0, 2}. Pick a randomN
fixed sequence b = {bi }∞i=1 ∈ {1, 2} . Consider the set of allβ-expansions)(∞Xai, ai ∈ Sbi
x |x =3ii=1K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

(a) b = 2, 2, 1, 2, 1, . . .

(b) b = 1, 2, 2, 2, 1, . . .

(c) b = 2, 2, 2, 1, 1, . . .

(d) b = 1, 1, 1, 2, 1, . . .

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Assuming the bi are choosen randomly, and with equal
probability, then the Hausdorff dimension of the support,almost surely, is log(6)log(9) .
Let µH be the Hausdorff measure with respect to thisdimension. Then the µH measure of this support, almost
surely, is 0.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Definition
We say that a iterated function system satisfies the Strong
Separation Condition if images under the maps neveroverlap.
We say that a iterated function system satisfies the Open
Set Condition if images under the maps never overlap
non-trivially.
We say that a iterated function system satisfies the Finite
Type Condition if images under the maps overlap in only
a finite number of ways (up to rescaling)

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Falconer 1986, Graf 1987: Determined their almost sure
Hausdorff dimension and measure properties.
As before, we can associate a measure µ with respect tob. These measures have a number of interesting almost
sure results.
Olsen 1994, Arbeiter Patzschke 1996: Considered themultifractal spectrum under the random analog of the open
set condition

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Theorem (Hare, H., Troscheit, to appear)
Consider a family of iterated function systems that satisfy theuniform strong separation conditions. Let µb be the measureconstructed by this family for particular b. There exists an α
and α such that for almost all b we have[α, α] = {dimloc µb (x) : x ∈ Support}= {dimloc µb (x) : x ∈ Support}= {dimloc µb (x) : x ∈ Support}.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Let v1 , v2 , v3 , . . . , vm be the finite number of ways thatimages of this map can overlap.
We construct the adjacency matrix Abi which determines
the number of vi at level n + 1 from the number of vi atlevel n where the digit choice is taken from Sbn .
Here we assume r is a common ratio of contraction.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Let v1 , v2 , v3 , . . . , vm be the finite number of ways thatimages of this map can overlap.
We construct the adjacency matrix Abi which determines
the number of vi at level n + 1 from the number of vi atlevel n where the digit choice is taken from Sbn .
Here we assume r is a common ratio of contraction.
Proposition (Hare, H., Troscheit, to appear)
Let the system of iterated function systems satisfy the finite
type condition. For almost all b, the box-counting dimension
and Hausdorff dimension of the support is equal tolog Ab1 . . . Abn.
n→∞n|log r |lim

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Fact
We can compute the local dimension of a point x with respect
to µb be a matrix product kB1 B2 B3 . . . Bn k1/n where the Bn
depends upon
Which vn we are in at each level
Which digit set Sbn
The location of x within vn at level n.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Fact
We can compute the local dimension of a point x with respect
to µb be a matrix product kB1 B2 B3 . . . Bn k1/n where the Bn
depends upon
Which vn we are in at each level
Which digit set Sbn
The location of x within vn at level n.
Definition
We say that the random iterated function system is commuting
if we have B1 . . . Bk1 , Bk1 +1 . . . Bk2 , etc are 1 × 1 matrices.

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Fact
We can compute the local dimension of a point x with respect
to µb be a matrix product kB1 B2 B3 . . . Bn k1/n where the Bn
depends upon
Which vn we are in at each level
Which digit set Sbn
The location of x within vn at level n.
Definition
We say that the random iterated function system is commuting
if we have B1 . . . Bk1 , Bk1 +1 . . . Bk2 , etc are 1 × 1 matrices.
Definition
We define Bj as the value of Bkj +1 . . . Bkj+1 and Nj the length of
this product.
K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Theorem (Hare, H., Troscheit, to appear)
Consider a random iterated function system of finite type andassume that the associated random self-similar measures, µb ,are regular. If the RIFS is commuting, then for almost all b, theset of attainable local dimensions for the measure µb is theclosed interval,#"
E log B1E (log B1 ),.
E(N1 ) log r E(N1 ) log r

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions

Thank you

K. E. Hare, K. G. Hare, S. Troscheit

Random homoeneous beta-expansions