Extensions of the random beta-transformation

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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

Transcript of Extensions of the random beta-transformation

Page 1: Extensions of the random beta-transformation

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

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1 Expansions in base β

2 The random β-transformation

3 Extensions and results

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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.

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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.

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FIGURE – Graphs of Tg and T`

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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.

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1 Expansions in base β

2 The random β-transformation

3 Extensions and results

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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.

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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?

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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.

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1 Expansions in base β

2 The random β-transformation

3 Extensions and results

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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 new dynamics

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The graph of transitions

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2

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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.

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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.

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The natural extension

• K not invertible.• Add the information of the past floors !• New system : the natural extension of the initial random β-transformation.

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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.

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Merci ! Thank you !

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