Combining aperiodic order with structural disorder: branching cellular automata

Transcription

1 Combining aperiodic order with structural disorder: branching cellular automata Michel Dekking Lorentz Center Workshop minutes lecture May 31, 2016

2 Branching cellular automata Branching cellular automata = Random substitutions with neighbor interaction First: Random substitutions without neighbor interaction: 2-D examples from FMD, Polymetric Brick Wall Patterns and Two-Dimensional Substitutions Journal of Integer Sequences, Vol. 16 (2013).

3 ion we start with a very simple self-similar example, which hen Random brick wall patterns isfy requirement [R3]. We take the 1 2 and 2 2 bricks with tion rules given by Take the 1 2 and 2 2 bricks with random substitution rules random : each expanded brick is replaced by one of the two possibilities with probability 1/2, independently of the other replacements.

4 Iterating 4 times Three realizations of σ 4 (B 22 ). Picked from a set of = elements! The set of all tilings with 16 rows with 1x2 and 2x2 rectangles has cardinality F

5 Multi-type branching processes Tools? The theory of multi-type branching processes can be applied to obtain further information on these patterns.

6 -type branching processes can be applied to obtain further info patterns. A pseudo-self-similar random substitution ext consider a pseudo-self-similar random substitution, where d with A coin a success is flippedprobability with a successp probability for someppfor some [0, 1]: p [0, 1]: 1 p 1 p p p maximal length of a vertical segment now is a random variab his pseudo-self-similar example the behavior of V max (p) as a fun ticularly interesting. Note that V max (0), but that V max (1) 8

7 Iterating 4 times Two realizations for p = 1/3 and p = 2/3 each: p = 1 3 p = 1 3 p = 2 3 p = 2 3

8 Revival: via entropy David Wing, Notions of Complexity in Substitution Dynamical Systems. PhD thesis, Oregon State University (2011). Johan Nilsson, On the Entropy of Random Fibonacci Words. arxiv: (2010). Johan Nilsson, On the Entropy of a Family of Random Substitutions, Monatsh. Math. 168 (2012). Johan Nilsson, On the Entropy of a Two Step Random Fibonacci Substitution, Entropy 15 (2013). R. Salgado-Garca and E. Ugalde, Exact Scaling in the Expansion-Modification System, J Stat Phys 153 (2013). Mathias Moll, On a Family of Random Noble Means Substitutions, Proceedings of ICQ12, Acta Physica Polonica A 126 (2014). D. Koslicki, M. Denker, Rocky Mountain J. Math. Substitution Markov chains and Martin boundaries (2016) Forthcoming.

9 Random substitutions What means random? Physicists: Mathematicians: random = a lot of freedom. random = (Ω, F, P).

10 Entropy of what? σ random substitution (X comb, T ), with T the shift. X comb = set of sequences whose finite subwords occur in some word from Supp(σ n (a)), for some letter a A. h comb = lim N 1 N log(#{w : w = N, w a,n Supp(σn (a))}).

11 FibMorse substitution: Example (from David Wings thesis): 0 01 { σ : 0 with probability p, 1 10 with probability 1 p. Let A n = Supp(σ n (0)), B n = Supp(σ n (1)). A 1 = {01}, B 1 = {10, 0} A 2 = {0110, 010}, B 2 = {1001, 001, 01} A 3 = { , , , , , 01001}.

12 Where is the randomness? For FibMorse: tree-indexed Bernoulli process. where P(σ = σ 1 ) = p, P(σ = σ 2 ) = 1 p, σ 1 (0) = 01, σ 1 (1) = 10, σ 2 (0) = 01, σ 2 (1) = 0. Ω : space of labelled binary trees, where the labels are σ 1, and σ 2. Words generated by the process: choose 0 or 1 at the root, and then level 1 consists of the words 01, 10 or 0Λ, where Λ is the empty word. (σ j (Λ) = ΛΛ)

13 Labelled trees σ 1 (0) = 01, σ 1 (1) = 10, σ 2 (0) = 01, σ 2 (1) = 0. σ 1 0 σ 2 σ σ 1 σ 1 σ 2 σ Λ

14 Random substitution dynamical systems Let X σ(ω) = set of sequences whose finite subwords occur in some σ(ω) n (a), for some letter a A. The random substitution is called tree primitive if there is an N such that for all letters a A and all labeled trees T N with N levels the N th level contains all letters from A. PROPOSITION Let σ be tree primitive. Then for almost all ω Ω (X σ(ω), T ) = (X comb, T ), and h((x σ(ω), T )) = h comb.

15 Examples The Godrèche-Luck random Fibonacci, the random noble mean substitutions FibMorse are all tree primitive. Let σ be the quarter-plane random substitution from Koslicki: { 10 with probability p, 0 σ : 01 with probability 1 p. 1 1 Then for almost all ω we (still!) have X σ(ω) = X comb.

16 History of random substitutions I Benoit Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) { σ : with probability 1 p, with probability p. Kahane, J.P., Peyrière, J., Sur Certaines Martingales de Benoit Mandelbrot, Adv. Math. 22 (1976). Almost sure Hausdorff dimension Peyrière, Jacques, Sur les colliers aléatoires de B. Mandelbrot, C. R. Acad. Sci. Paris Sér. A-B 286, (1978). Random rivers! Peyrière Jacques, Processus de naissance avec interaction des voisins, évolution de graphes. Ann. l institut Fourier 31, (1981).

17 Mandelbrot percolation = fractal percolation Benoit Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974),

18 History of random substitutions II Benoit Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974). Mandelbrot introduced this model as a critique of Kolmogorov s model of turbulence, but it is only phenomenological Chayes, J. T., L. Chayes, and R. Durrett (1988) Connectivity properties of Mandelbrots percolation process, Probab.Theory Related Fields 77. Dekking, F. M. and R. W. J. Meester. (1990). On the structure of Mandelbrots percolation process and other random Cantor sets,j. Statist. Phys. 58.

19 Neighbour interaction I Benoit Mandelbrot, Intermittent turbulence... (1974). Siebesma, A. P. and Tremblay, R. R. and Erzan, A. and Pietronero, L., Multifractal cascades with interactions, Physica A. Europhysics Journal 156, (1989). Only 1-dimensional! 0 00 σ : 01 with probability 1 2 (1 p(z)), 1 10 with probability 1 2 (1 p(z)), 11 with probability p(z). where p(z) is a number in [0, 1] for z = 0, 1, 2 and z is equal to the number neighbours of the 1 that is replaced by σ(1).

20 Neighbour interaction II Benoit Mandelbrot, Intermittent turbulence... (1974). F.M. Dekking and P. van der Wal,...BCA (2001)

21 Neighbour interaction finds boundaries F.M. Dekking and P. van der Wal,...BCA (2001) A set of dimension 2. Its boundary

22 Neighbour interaction III Order 6 and 7 of Coffey-Hyman example.