Post on 19-Aug-2020
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
Random systems Results
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
Random systems Results
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))
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Random systems Results
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
Random systems Results
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
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
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
Random systems Examples
Example 1: random β-transformations
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
Random systems Examples
Example 1: random β-transformations
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
Random systems Examples
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]
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Random systems Examples
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
Random systems Examples
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
Random systems Examples
Thank you!
M. Maggioni 20 / 22
Random systems Examples
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M. Maggioni 21 / 22
Random systems Examples
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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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M. Maggioni 22 / 22