Invariant densities for random systems
Marta Maggioni
joint work with Charlene Kalle
Universiteit Leiden

Numeration2018
May 23, 2018

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Random systems

Setting(Ω ⊆ N, p) prob space
T := {Tj : X → X, j ∈ Ω} family of maps
T is a random system of the space X of probability p, if
T (x) := Tj (x) with probability pj

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Random systems

Motivation

Stochastic perturbations
Particles systems
Number expansions
β, Lüroth, dyadic expansions, etc.

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Random systems

MotivationRandom β-transformations [DK03]

T0 (x) =(βx,βx − 1,(1]if x ∈ [0, β(β−1)
11if x ∈ ( β(β−1), β−1]

1β−1,T1 (x) =

1β−1

1β−1

1

12−β
β−12−ββ−1011β(β−1) β−1(a) T0

M. Maggioni

if x ∈ [0, β1 )1if x ∈ [ β1 , β−1]βx,
βx − 1,01β(b) T1

1
β−101
β−1(c) T

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Random systems

ACIM: definition(X, B, µp , T, p) random systemZ
ACIM:µp (A) =h dλ =AXpj µp (Tj−1 A)

for all A ∈ Bj∈Ω

Perron-Frobenius operator
RA PT h dλ =PT h =

PRT −1 (A) h dλj∈Ω PTj pj h

PT h = h

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ACIM µh

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Random systems

Existing formulas

Lasota-Yorke linear maps
Same slopes
Parry, Dajani, Kempton, Suzuki for the β-transformations(deterministic and random)

Different slopes
Kopf [Kop90], Góra (deterministic)
Our approach (random)

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Random systems

Results

Setting
T = {Tj : [0, 1] → [0, 1], j ∈ Ω} expanding on average wrt psupXx∈[0,1] j∈Ωpj<1|Tj0 (x)|Tj piecewise linear{I1 , ..., IN } partition for the set {0 = z0 < z1 < ... < zN = 1}
discontinuity points
Ti,j (x) = ki,j x + di,j(Thm, [Ino12]) T admits an ACIM

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Random systems

Results

Assumptions

1. T (0), T (1) ∈ {0, 1}
2. for every i there exists n:Ppjj∈Ω ki,j di,jP
pj
1 − j∈Ω ki,jM. Maggioni

pjj∈Ω kn,j dn,jP−pj
1 − j∈Ω kn,jP

6= 0

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Random systems

Results

Definitionsω ∈ ΩN pathy ∈ [0, 1] pointt ∈ N instant of timen ∈ {1, . . . , N } intervalpωtτω (y, t) := ki,ω,tδω (y, t) :=Qt

KIn (y) :=P

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if Tωt−1 (y) ∈ Ii
1n=0 τω (y, n)t≥1Pω∈Ωt δω (y, t)1In (Tω1t−1 (y))

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Random systems

Results

ResultsThm. (Kalle, M., to appear)
Under the previous assumptions,hγ (x) = cN−1X

γmm=1X  pll∈Ωkm,lLam,l (x) −plkm+1,lLbm,l (x)is a T -invariant function.
am,l = km,l zm + dm,l ,Ly (x) =X X

bm,l = km+1,l zm + dm+1,lδω (y, t)1[0,Tω (y)) (x)t≥0 ω∈Ωt

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Random systems

Results

Results

Procedure:T→M→

Mγ = 0→

hγfor

M=X
j∈Ω

M. Maggionipjpj
KIn (ai,j ) −KIn (bi,j ) + qn,iki,jki+1,jn,i

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Random systems

Results

Results

Not straightforward
There always exists γ 6= 0 (ass. 2.)
hγ is T -invariant (ass. 1.){I1 , . . . , IN } arbitrary (endpoints and size of this set)

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Random systems

Results

ResultsThm. (Kalle, M., to appear)For Ω finite and T expanding, the construction gives all possible
T -invariant densities.
Idea:

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MM̂U†

γγ̂U†

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Random systems

ExamplesExample 1: random β-transformations[DdV07]: special β[Kem14]: h for all 1 < β < 2, unbiased caseh(x) = c∞X1(2β)nn=0Xω1 ···ωn ∈{0,1}n
1[0,Rn1 ] (x)(1)] (x) + 1[Rn( 2−β ), β−1
β,ω1 ···ωnβ,ω ···ωn β−1
1[Suz17]: h for all 1 < β < 2, biased cases

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Random systems

ExamplesExample 1: random β-transformations
h for all 1 < β < 2 for p0 = p1 = 121β−112−ββ−10hγ (x) = kX Xt≥0 ω∈{0,1}

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1β−1
11(x) + 1[Tω ( 2−β ), 1 ] (x)β−1 β−1(2β)t [0,Tω (1))
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Random systems

ExamplesExample 1: random β-transformations

KIn (1)

KInc1c2c32−β β−1

KIn (0)KIn

1β−1c3c2
c1001 β−100
1β−1

↓
1β


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1
1+ 2β(c1 − β−1)
1− β1 + 2β
c21
c2β 3

1− 2βc3  1=0
 11
1− β1 − 2β(c1 − β−1)1
1− 2βc2β16 / 22

Random systems

ExamplesExample 2: random (α, β)-transformations
T0 (x) =(βx,α(βx − 1),βif x ∈ [0, 1/β)
if x ∈ [1/β, 1]

andT1 (x) =(βx,βx − 1,if x ∈ [0, 1/β)
if x ∈ [1/β, 1]

β1
1β1β3

0 13 12 1ββ

β 2 = β + 1, α = 1/β→β1βhγ = c((β 2 p + β)1A + (p + β)1B + β1C + 11D )[DHK09]M. Maggioni

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Random systems

ExamplesExample 3: random Lüroth map with bounded digits
x=X(−1)sn−1 (rn + ωn − 1)

n≥11

k=1

10 111 1
54 3 21nY1rk (rk − 1)
10 111 1

154 3 2TL (x) := n(n − 1)x − (n − 1)

and01 13 2

1TA (x) := 1 − TL (x)[Lür83, BBDK94, BI09, LY78, Pel84]
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Random systems

ExamplesExample 3: random Lüroth map with bounded digits

1123231231 13 201 13 21I1(g) T00I1131(h) T11(i) Thγ (x) = 3/8(3 · 1[ 1 , 2 ] (x) + 5 · 1( 2 ,1] (x))
3 33digits frequency: 2 → 13/16, 3 → 3/16

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Random systems

ExamplesThank you!

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Random systems

ExamplesJ. Barrionuevo, R. M. Burton, K. Dajani, and C. Kraaikamp.
Ergodic properties of generalized Luroth series.
TU Delft Report, 94-105:1–16, 1994.
L. Barreira and G. Iommi.
Frequency of digits in the Lüroth expansion.
J. Number Theory, 129(6):1479–1490, 2009.
K. Dajani and M. de Vries.
Invariant densities for random β-expansions.
J. Eur. Math. Soc., 9(1):157–176, 2007.
K. Dajani, Y. Hartono, and C. Kraaikamp.
Mixing properties of (α, β)-expansions.
Ergodic Theory Dynam. Systems, 29(4):1119–1140, 2009.
K. Dajani and C. Kraaikamp.
Random β-expansions.
Ergodic Theory Dynam. Systems, 23(2):461–479, 2003.
T. Inoue.
Invariant measures for position dependent random maps with continuous random parameters.
Studia Math., 208(1):11–29, 2012.
K. Kempton.
On the invariant density of the random β-transformation.
Acta Math. Hungar., 142(2):403–419, 2014.
C. Kopf.
Invariant measures for piecewise linear transformations of the interval.
Appl. Math. Comput., 39(2, part II):123–144, 1990.

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Random systems

ExamplesJ. Lüroth.
Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe.
Math. Ann., 21(3):411–423, 1883.
T. Y. Li and J. A. Yorke.
Ergodic transformations from an interval into itself.
Trans. Amer. Math. Soc., 235:183–192, 1978.
S. Pelikan.
Invariant densities for random maps of the interval.
Trans. Amer. Math. Soc., 281(2):813–825, 1984.
S. Suzuki.
Invariant density functions of random β-transformations.
Ergodic Theory and Dynamical Systems, page 122, 2017.

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