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 = { f 1 , . . . , f N : 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 = { f 1 , . . . , f N : 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 f i ( C ) = C , (ii) C 1 = C 1 ⊂ C obeying (i) ⇒ C 1 = C . K Le´ sniak (N.Copernicus University) Random iteration March 2014 2 / 6

Environment IFS X – complete metric space F = { f 1 , . . . , f N : 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 f i ( C ) = C , (ii) C 1 = C 1 ⊂ C obeying (i) ⇒ C 1 = 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 orbit Given starting point x 0 ∈ X and the sequence of symbols ( driver ) i 1 , i 2 , . . . ∈ { 1 , . . . , N } we build an orbit x n := f i n ( x n − 1 ), n ≥ 1, and the corresponding ω (( x n )) := � ∞ omega-limit set m =0 { x n : n ≥ m } K Le´ sniak (N.Copernicus University) Random iteration March 2014 3 / 6

Random orbit orbit Given starting point x 0 ∈ X and the sequence of symbols ( driver ) i 1 , i 2 , . . . ∈ { 1 , . . . , N } we build an orbit x n := f i n ( x n − 1 ), n ≥ 1, and the corresponding ω (( x n )) := � ∞ omega-limit set m =0 { x n : n ≥ m } random i 1 i 2 i 3 . . . ∈ { 1 , . . . , N } ∞ – disjunctive , if ∀ τ 0 τ 1 τ 2 ...τ k ∃ n s.t. i n i n +1 i n +2 ... i n + k = τ 0 τ 1 τ 2 ...τ k K Le´ sniak (N.Copernicus University) Random iteration March 2014 3 / 6

Chaos Game Representation F = { f i : X → X , i = 1 , . . . , N } Assumptions: (D) the driver of the orbit ( x n ) ∞ n =1 is disjunctive, (B) F admits a nonempty closed bounded minimal invariant set, (CO) F is ‘contractive on orbits’, i.e., there exists ( u 1 , . . ., u k ) ∈ { 1 , . . ., N } k s.t. f u k ◦ . . . ◦ f u 1 is a Lipschitz contraction when restricted to the branching tree � { f i m ◦ . . . ◦ f i 1 ( x 0 ) } . i 1 ,..., i m ∈{ 1 ,..., N } m ≥ 0 K Le´ sniak (N.Copernicus University) Random iteration March 2014 4 / 6

Chaos Game Representation F = { f i : X → X , i = 1 , . . . , N } Assumptions: (D) the driver of the orbit ( x n ) ∞ n =1 is disjunctive, (B) F admits a nonempty closed bounded minimal invariant set, (CO) F is ‘contractive on orbits’, i.e., there exists ( u 1 , . . ., u k ) ∈ { 1 , . . ., N } k s.t. f u k ◦ . . . ◦ f u 1 is a Lipschitz contraction when restricted to the branching tree � { f i m ◦ . . . ◦ f i 1 ( x 0 ) } . i 1 ,..., i m ∈{ 1 ,..., N } m ≥ 0 Conclusion: ω (( x n )) 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 ( Z n ) n ≥ 1 with values in the alphabet { 1 , .., N } is called disjunctive , if every finite word appears a.s. in its outcome. Remark ( Z n ) 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 ( Z n ) 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 ∀ σ 1 , . . ., σ n ∈ { 1 , .., N } Pr( Z n = σ n | Z n − 1 = σ n − 1 , . . ., Z 1 = σ 1 ) ≥ α n , (De) ∀ m ≥ 1 ∞ m � � α ( n − 1)+ l = ∞ . n =1 l =1 Conclusion: ( Z n ) n ≥ 1 is disjunctive. K Le´ sniak (N.Copernicus University) Random iteration March 2014 5 / 6

Disjunctive stochastic prosesses A process ( Z n ) 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) α n ≡ α > 0, encompasses homogeneous Bernoulli schemes and Markov chains, 2 ∃ b > 0 α − 1 ≈ (ln n ) b , n 3 ∀ c > 0 α − 1 ≪ n c . n 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