Marta Maggioni Charlene Kalle - Sciencesconf.org · Invariant densities for random systems Marta...

Invariant densities for random systems

Marta Maggioni

joint work with Charlene Kalle

Universiteit Leiden

Numeration2018

May 23, 2018

M. Maggioni 1 / 22

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

M. Maggioni 2 / 22

Random systems

Motivation

Stochastic perturbations

Particles systems

Number expansions

β, Luroth, dyadic expansions, etc.

M. Maggioni 3 / 22

Random systems

Motivation

Random β-transformations [DK03]

T0(x) =

βx, if x ∈ [0, 1

β(β−1)]

βx− 1, if x ∈ ( 1β(β−1)

, 1β−1

], T1(x) =

βx, if x ∈ [0, 1

β)

βx− 1, if x ∈ [ 1β, 1β−1

]

0 1β(β−1)

1β−1

1β−1

2−ββ−1

(a) T0

0 1β

1β−1

1β−1

1

(b) T1

0 1β−1

1β−1

12−ββ−1

(c) T

M. Maggioni 4 / 22

Random systems

ACIM: definition

(X,B, µp, T, p) random system

ACIM: µp(A) =

∫Ah dλ =

∑j∈Ω

pjµp(T−1j A) for all A ∈ B

Perron-Frobenius operator∫A PTh dλ =

∫T−1(A) h dλ

PTh =∑

j∈Ω PTjpjh

PTh = h ACIM µh

M. Maggioni 5 / 22

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], Gora (deterministic)Our approach (random)

M. Maggioni 6 / 22

Random systems Results

Setting

T = Tj : [0, 1]→ [0, 1], j ∈ Ω expanding on average wrt p

supx∈[0,1]

∑j∈Ω

pj|T ′j(x)|

< 1

Tj piecewise linear

I1, ..., IN partition for the set 0 = z0 < z1 < ... < zN = 1discontinuity points

Ti,j(x) = ki,jx+ di,j

(Thm, [Ino12]) T admits an ACIM

M. Maggioni 7 / 22


Assumptions

1. T (0), T (1) ∈ 0, 1

2. for every i there exists n:∑j∈Ω

pjki,j

di,j

1−∑

j∈Ωpjki,j

−∑

j∈Ωpjkn,j

dn,j

1−∑

j∈Ωpjkn,j

6= 0

M. Maggioni 8 / 22


Definitions

ω ∈ ΩN path

y ∈ [0, 1] point

t ∈ N instant of time

n ∈ 1, . . . , N interval

τω(y, t) :=pωtki,ωt

, if Tωt−11

(y) ∈ Ii

δω(y, t) :=∏tn=0 τω(y, n)

KIn(y) :=∑

t≥1

∑ω∈Ωt δω(y, t)1In(Tωt−1

1(y))

M. Maggioni 9 / 22


Results

Thm. (Kalle, M., to appear)

Under the previous assumptions,

hγ(x) = cN−1∑m=1

γm∑l∈Ω

[plkm,l

Lam,l(x)− plkm+1,l

Lbm,l(x)

]is a T -invariant function.

am,l = km,lzm + dm,l, bm,l = km+1,lzm + dm+1,l

Ly(x) =∑t≥0

∑ω∈Ωt

δω(y, t)1[0,Tω(y))(x)

M. Maggioni 10 / 22


Results

Procedure:

T → M → Mγ = 0 → hγ

for

M =

(∑j∈Ω

[pjki,j

KIn(ai,j)−pj

ki+1,jKIn(bi,j)

]+ qn,i

)n,i

M. Maggioni 11 / 22


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)

M. Maggioni 12 / 22


Results

Thm. (Kalle, M., to appear)

For Ω finite and T expanding, the construction gives all possibleT -invariant densities.

Idea:

M M †U

γ γ†U

M. Maggioni 13 / 22

Random systems Examples

Example 1: random β-transformations

[DdV07]: special β

[Kem14]: h for all 1 < β < 2, unbiased case

h(x) = c∞∑n=0

1

(2β)n

( ∑ω1···ωn∈0,1n

1[0,Rnβ,ω1···ωn

(1)](x) + 1[Rnβ,ω1···ωn

( 2−ββ−1

), 1β−1

](x)

)

[Suz17]: h for all 1 < β < 2, biased cases

M. Maggioni 14 / 22



h for all 1 < β < 2 for p0 = p1 = 12

0 1β−1

1β−1

1

2−ββ−1

hγ(x) = k∑t≥0

∑ω∈0,1t

1

(2β)t

(1[0,Tω(1))(x) + 1

[Tω( 2−ββ−1

), 1β−1

](x)

)

M. Maggioni 15 / 22



KIn(1) KIn( 2−ββ−1

)KIn(0) KIn

( 1β−1

)c1 c3

1β−1

0

c2 c2 0 0

c3 c1 0 1β−1

↓

1β

+ 12β

(c1 − 1β−1

) − 12βc3

− 1β

+ 12βc2

1β− 1

2βc2

12βc3 − 1

β− 1

2β(c1 − 1

β−1)

(

11

)= 0

M. Maggioni 16 / 22


Example 2: random (α, β)-transformations

T0(x) =

βx, if x ∈ [0, 1/β)αβ

(βx− 1), if x ∈ [1/β, 1]and T1(x) =

βx, if x ∈ [0, 1/β)

βx− 1, if x ∈ [1/β, 1]

0 1β3

1β2

1β

1

1

β

1β

1β3

β

β2 = β + 1, α = 1/β → hγ = c((β2p+ β)1A + (p+ β)1B + β1C + 11D)

[DHK09]

M. Maggioni 17 / 22


Example 3: random Luroth map with bounded digits

x =∑n≥1

(−1)sn−1(rn + ωn − 1)

n∏k=1

1

rk(rk − 1)

0 151413

12

1

1

0 151413

12

1

1

0 13

12

1

1

TL(x) := n(n− 1)x− (n− 1) and TA(x) := 1− TL(x)

[Lur83, BBDK94, BI09, LY78, Pel84]

M. Maggioni 18 / 22


Example 3: random Luroth map with bounded digits

0 1

1

13

12

23

I1

(g) T0

0 1

1

13

12

23

I1

(h) T1

13 1

1

23

(i) T

hγ(x) = 3/8(3 · 1[ 13, 23

](x) + 5 · 1( 23,1](x))

digits frequency: 2→ 13/16, 3→ 3/16

M. Maggioni 19 / 22


Thank you!

M. Maggioni 20 / 22


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M. Maggioni 21 / 22


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Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe.Math. Ann., 21(3):411–423, 1883.

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Ergodic transformations from an interval into itself.Trans. Amer. Math. Soc., 235:183–192, 1978.

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M. Maggioni 22 / 22

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