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Extensions of the random beta-transformation
Younès Tierce
Université de Rouen-Normandie, LMRS, supervised by Thierry de la Rue and Jean-Baptiste Bardet
November 9, 2021
Younès Tierce Extensions of the random beta-transformation November 9, 2021 1 / 31
1 Expansions in base β
2 The random β-transformation
3 Extensions and results
Younès Tierce Extensions of the random beta-transformation November 9, 2021 2 / 31
Expansions in base β
• β > 1
• x ∈ Iβ :=[
0;bβc
β−1
]
Expansions in base β
x =+∞
∑n=1
xn
βn , xn ∈ 0, . . . ,bβc.
We will now fix 1 < β < 2.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 3 / 31
How to obtain an expansion in base β ?
The greedy map (Rényi, 1957)
Tg :
Iβ → Iβ
x 7→
βx if x < 1
β
βx−1 otherwise
.
Provides the greatest expansion of a given number in the lexicographic order (Erdös,Joo).
The lazy map
T` :
Iβ → Iβ
x 7→
βx if x 6 1
β(β−1)
βx−1 otherwise
.
Provides the smallest expansion of a given number in the lexicographic order.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 4 / 31
FIGURE – Graphs of Tg and T`
Younès Tierce Extensions of the random beta-transformation November 9, 2021 5 / 31
Ergodic properties of the associated systems
Rényi (1957) and Parry (1960)The greedy map Tg has a unique absolutely continuous invariant probability measureνβ, whose density is proportional to the function
+∞
∑n=0
1βn 1[0,Tn
g (1)](x).
Smorodinsky (1973)The natural extension of the system (Iβ,νβ,Tg) is a Bernoulli shift.
Dajani and Kraaikamp (2002)The systems associated to the greedy and lazy maps are isomorphic.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 6 / 31
1 Expansions in base β
2 The random β-transformation
3 Extensions and results
Younès Tierce Extensions of the random beta-transformation November 9, 2021 7 / 31
Generate every expansions in base β ?
• Idea : choose between greedy and lazy at each step.• Formally : Ω = g, `N, with the Bernoulli measure mp.
Random dynamical system
Kβ :Ω× Iβ → Ω× Iβ
(ω,x) 7→ (σ(ω),Tω0(x)).
where σ is the left shift.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 8 / 31
Questions
• Existence of an absolutely continuous probability measure µp on Iβ such thatmp⊗µp is Kβ-invariant? Uniqueness?
• Explicit expression of its density?• Ergodic properties of the associated system?
Younès Tierce Extensions of the random beta-transformation November 9, 2021 9 / 31
How to get such a measure?
• Idea : Construct an extension of Kβ with a “simple” invariant measure, thenproject this measure on Iβ.
• 2014 : extension in the case p = 12 by Kempton.
→ Expression of the density of µ1/2.• 2019 : Generalised expression with transfer operators by Suzuki.→ µp for any p.
• Our work : extension of the system for any p.→ Provides an expression of µp, and ergodic properties of the randomβ-transformation.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 10 / 31
1 Expansions in base β
2 The random β-transformation
3 Extensions and results
Younès Tierce Extensions of the random beta-transformation November 9, 2021 11 / 31
Younès Tierce Extensions of the random beta-transformation November 9, 2021 12 / 31
The new dynamics
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The new dynamics
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The new dynamics
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The new dynamics
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The new dynamics
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The graph of transitions
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2
3
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Density of µp (T.)
ρp(x) =+∞
∑n=0
1βn
(Cg ∑
ω1,...,ωn∈g,`nmp([ω1, ...,ωn]
n−10 )1[0,Tω1 ,...,ωn (1+)](x)
+C` ∑ω1,...,ωn∈g,`n
mp([ω1, ...,ωn]n−10 )1[Tω1 ,...,ωn (s(1)−);
1β−1 ]
(x)
).
where :• Cg and C` are two positive explicit constants,
• mp([ω1, ...,ωn]n−10 ) is the Bernoulli measure of the cylinder [ω1, ...,ωn]
n−10 .
Similar form of the density previously obtained by Suzuki.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 27 / 31
The induced system on Eg
• Induced transformation of K on the greedy base Eg : look at the returns to thebase Eg.
• Partition Eg according to the encountered floors until the next return to Eg.
→ Provides a sequence of independent partitions.→ Generates the induced system.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 28 / 31
The natural extension
• K not invertible.• Add the information of the past floors !• New system : the natural extension of the initial random β-transformation.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 29 / 31
Results
• Ergodicity of the natural extension⇒ Ergodicity of the initial random dynamicalsystem + uniqueness of the measure µp as an invariant absolutely continuousprobability measure.
• Consequence : for mp⊗Leb a.e (ω,x) ∈Ω× Iβ,
1N
N−1
∑n=0
δπ(Knβ(ω,x))→ µp.
Bernoulli system (T.)The natural extension of the system (Ω× Iβ,mp⊗µp,Kβ) is a Bernoulli shift.
Younès Tierce Extensions of the random beta-transformation November 9, 2021 30 / 31
Merci ! Thank you !
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