Random iteration and disjunctive processes Krzysztof Le sniak - - PowerPoint PPT Presentation

Random iteration and disjunctive processes

Krzysztof Le´ sniak

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toru´ n

March 2014

K Le´ sniak (N.Copernicus University) Random iteration March 2014 1 / 6

Environment

IFS X – complete metric space F = {f1, . . . , fN : X → X} system of nonexpansive (1-Lipschitz) functions

K Le´ sniak (N.Copernicus University) Random iteration March 2014 2 / 6

Environment

IFS X – complete metric space F = {f1, . . . , fN : X → X} system of nonexpansive (1-Lipschitz) functions Invariance A nonempty closed bounded set C ⊂ X is minimal invariant for F, when (i) F(C) := N

i=1 fi(C) = C,

(ii) C1 = C1 ⊂ C

• beying (i) ⇒ C1 = C.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 2 / 6

Environment

IFS X – complete metric space F = {f1, . . . , fN : X → X} system of nonexpansive (1-Lipschitz) functions Invariance A nonempty closed bounded set C ⊂ X is minimal invariant for F, when (i) F(C) := N

i=1 fi(C) = C,

(ii) C1 = C1 ⊂ C

• beying (i) ⇒ C1 = C.

Example (Kaczmarz) C = {x∗} – common fixed point for a system of orthogonal projections

K Le´ sniak (N.Copernicus University) Random iteration March 2014 2 / 6

Random orbit

• rbit

Given starting point x0 ∈ X and the sequence of symbols (driver) i1, i2, . . . ∈ {1, . . . , N} we build an orbit xn := fin(xn−1), n ≥ 1, and the corresponding

• mega-limit set

ω((xn)) := ∞

m=0 {xn : n ≥ m}

K Le´ sniak (N.Copernicus University) Random iteration March 2014 3 / 6

Random orbit

• rbit

Given starting point x0 ∈ X and the sequence of symbols (driver) i1, i2, . . . ∈ {1, . . . , N} we build an orbit xn := fin(xn−1), n ≥ 1, and the corresponding

• mega-limit set

ω((xn)) := ∞

m=0 {xn : n ≥ m}

random i1i2i3 . . . ∈ {1, . . . , N}∞ – disjunctive, if ∀ τ0τ1τ2...τk ∃ n s.t. inin+1in+2...in+k = τ0τ1τ2...τk

K Le´ sniak (N.Copernicus University) Random iteration March 2014 3 / 6

Chaos Game Representation

F = {fi : X → X, i = 1, . . . , N} Assumptions: (D) the driver of the orbit (xn)∞

n=1 is disjunctive,

(B) F admits a nonempty closed bounded minimal invariant set, (CO) F is ‘contractive on orbits’, i.e., there exists (u1, . . ., uk) ∈ {1, . . ., N}k s.t. fuk ◦ . . . ◦ fu1 is a Lipschitz contraction when restricted to the branching tree

• i1,...,im∈{1,...,N}

m≥0

{fim◦. . .◦fi1(x0)}.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 4 / 6

Chaos Game Representation

F = {fi : X → X, i = 1, . . . , N} Assumptions: (D) the driver of the orbit (xn)∞

n=1 is disjunctive,

(B) F admits a nonempty closed bounded minimal invariant set, (CO) F is ‘contractive on orbits’, i.e., there exists (u1, . . ., uk) ∈ {1, . . ., N}k s.t. fuk ◦ . . . ◦ fu1 is a Lipschitz contraction when restricted to the branching tree

• i1,...,im∈{1,...,N}

m≥0

{fim◦. . .◦fi1(x0)}. Conclusion: ω((xn)) is a closed minimal invariant set for F.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 4 / 6

Disjunctive stochastic prosesses

A process (Zn)n≥1 with values in the alphabet {1, .., N} is called disjunctive, if every finite word appears a.s. in its outcome. Remark (Zn)n≥1 is not necessarily stationary by any means.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 5 / 6

Disjunctive stochastic prosesses

A process (Zn)n≥1 with values in the alphabet {1, .., N} is called disjunctive, if every finite word appears a.s. in its outcome. Assumptions: (B-V) ∀n ≥ 1 ∀σ1, . . ., σn ∈ {1, .., N} Pr(Zn = σn | Zn−1 = σn−1, . . ., Z1 = σ1) ≥ αn, (De) ∀m ≥ 1

∞

• n=1

m

• l=1

α(n−1)+l = ∞. Conclusion: (Zn)n≥1 is disjunctive.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 5 / 6

Disjunctive stochastic prosesses

A process (Zn)n≥1 with values in the alphabet {1, .., N} is called disjunctive, if every finite word appears a.s. in its outcome. Example

1 (Barnsley & Vince 2011) αn ≡ α > 0, encompasses homogeneous

Bernoulli schemes and Markov chains,

2 ∃b > 0 α−1

n

≈ (ln n)b,

3 ∀c > 0 α−1

n

≪ nc.

K Le´ sniak (N.Copernicus University) Random iteration March 2014 5 / 6

KL thx

arXiv:1307.5920 (influenced by M. Barnsley); PODE Bia lystok 2013 presented; work in progress ,,On discrete stochastic processes with disjunctive outcomes” (influenced by ¨

• O. Stenflo); Bull. Aust. Math Soc. – accepted

THANK YOU!

K Le´ sniak (N.Copernicus University) Random iteration March 2014 6 / 6