The Dynamics Group of Asynchronous Systems Henning S. Mortveit - PowerPoint PPT Presentation

Background and Terminology Update Sequence Independent SDS Closing Remarks and Outlook The Dynamics Group of Asynchronous Systems Henning S. Mortveit Department of Mathematics & NDSSL, Virginia Bioinformatics Institute Dedicated to Eric Goles for his 60 th birthday! DISCO 2011 Instituto de Sistemas Complejos de Valpara´ ıso November 24–26, 2011, Valpara´ ıso, Chile

Background and Terminology Update Sequence Independent SDS Closing Remarks and Outlook Talk Overview – Dynamics Groups ◮ Using tools from group theory to assess long-term dynamics of asynchronous discrete dynamical systems. The notion of update sequence independence. The dynamics group of an update sequence independent system. Relations to Coxeter theory and Coxeter groups. Outlook and open questions.

Background and Terminology Sequential Dynamical Systems Update Sequence Independent SDS SDS Example Closing Remarks and Outlook Sequential Dynamical Systems (SDS) ◮ A subclass of graph dynamical systems (GDS). Constructed from: A (dependency) graph X with vertex set v [ X ] = { 1 , 2 , . . . , n } . 3 n [4]=(3,4,5,8) 1 For each vertex v a state x v ∈ K (e.g. K = F 2 = { 0 , 1 } ) and an X-local function F v : K n − 4 f 4 ( x 3 , x 4 , x 5 , x 8 ) → K n 5 8 F v ( x 1 , x 2 , . . . , x n ) = ( x 1 , . . . , f v ( x [ v ]) , , . . . , x n ) . 2 | {z } 7 6 vertex function v A word w = w 1 w 2 · · · w k over the vertex set of X .

Background and Terminology Sequential Dynamical Systems Update Sequence Independent SDS SDS Example Closing Remarks and Outlook Sequential Dynamical Systems (SDS) ◮ A subclass of graph dynamical systems (GDS). Constructed from: A (dependency) graph X with vertex set v [ X ] = { 1 , 2 , . . . , n } . 3 n [4]=(3,4,5,8) 1 For each vertex v a state x v ∈ K (e.g. K = F 2 = { 0 , 1 } ) and an X-local function F v : K n − 4 f 4 ( x 3 , x 4 , x 5 , x 8 ) → K n 5 8 F v ( x 1 , x 2 , . . . , x n ) = ( x 1 , . . . , f v ( x [ v ]) , , . . . , x n ) . 2 | {z } 7 6 vertex function v A word w = w 1 w 2 · · · w k over the vertex set of X . ◮ The SDS map F w : K n − → K n is: F w = F w ( k ) ◦ F π ( k − 1) ◦ · · · ◦ F w (1)

Background and Terminology Sequential Dynamical Systems Update Sequence Independent SDS SDS Example Closing Remarks and Outlook SDS – An example ◮ System components: Circle graph on 4 vertices: X = Circle 4 1 F Update sequence: π = (1 , 2 , 3 , 4) 1 Vertex functions: F 2 4 nor 3 ( x 1 , x 2 , x 3 ) = (1 + x 1 )(1 + x 2 )(1 + x 3 ) F 4 2 The X -local map for vertex 1: F 3 3 F 1 ( x 1 , x 2 , x 3 , x 4 ) = ( nor 3 ( x 1 , x 2 , x 4 ) , x 2 , x 3 , x 4 ) Dependency graph

Background and Terminology Sequential Dynamical Systems Update Sequence Independent SDS SDS Example Closing Remarks and Outlook SDS – An example ◮ System components: Circle graph on 4 vertices: X = Circle 4 1 F Update sequence: π = (1 , 2 , 3 , 4) 1 Vertex functions: F 2 4 nor 3 ( x 1 , x 2 , x 3 ) = (1 + x 1 )(1 + x 2 )(1 + x 3 ) F 4 2 The X -local map for vertex 1: F 3 3 F 1 ( x 1 , x 2 , x 3 , x 4 ) = ( nor 3 ( x 1 , x 2 , x 4 ) , x 2 , x 3 , x 4 ) Dependency graph ◮ System update: F 1 ( x 1 , x 2 , x 3 , x 4 ) = (0 , 0 , 0 , 0) �→ (1 , 0 , 0 , 0) and F 2 F 3 F 4 (1 , 0 , 0 , 0) �→ (1 , 0 , 0 , 0) �→ (1 , 0 , 1 , 0) �→ (1 , 0 , 1 , 0)

Background and Terminology Sequential Dynamical Systems Update Sequence Independent SDS SDS Example Closing Remarks and Outlook SDS – An example ◮ System components: Circle graph on 4 vertices: X = Circle 4 1 F Update sequence: π = (1 , 2 , 3 , 4) 1 Vertex functions: F 2 4 nor 3 ( x 1 , x 2 , x 3 ) = (1 + x 1 )(1 + x 2 )(1 + x 3 ) F 4 2 The X -local map for vertex 1: F 3 3 F 1 ( x 1 , x 2 , x 3 , x 4 ) = ( nor 3 ( x 1 , x 2 , x 4 ) , x 2 , x 3 , x 4 ) Dependency graph ◮ System update: 1000 (1234) 1100 0101 0010 0011 F 1 ( x 1 , x 2 , x 3 , x 4 ) = (0 , 0 , 0 , 0) �→ (1 , 0 , 0 , 0) and 0111 0000 0100 1001 1011 1101 F 2 F 3 F 4 1111 1010 (1 , 0 , 0 , 0) �→ (1 , 0 , 0 , 0) �→ (1 , 0 , 1 , 0) �→ (1 , 0 , 1 , 0) 0001 0110 1110 ◮ SDS map: Phase space F π (0 , 0 , 0 , 0) = (1 , 0 , 1 , 0)

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Definition (Update sequence independence) i =1 of X -local maps over a finite state space K n are word (resp. A sequence F = ( F i ) n permutation) update sequence independent, if there exists P ⊂ K n such that for all fair words w ∈ W ′ X (resp. w ∈ S X ) we have Per ( F w ) = P .

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Definition (Update sequence independence) i =1 of X -local maps over a finite state space K n are word (resp. A sequence F = ( F i ) n permutation) update sequence independent, if there exists P ⊂ K n such that for all fair words w ∈ W ′ X (resp. w ∈ S X ) we have Per ( F w ) = P . ◮ We usually just say that F = ( F i ) i is π -independent or w -independent. ◮ Clearly, word independence implies permutation independence; the converse is false.

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Definition (Update sequence independence) i =1 of X -local maps over a finite state space K n are word (resp. A sequence F = ( F i ) n permutation) update sequence independent, if there exists P ⊂ K n such that for all fair words w ∈ W ′ X (resp. w ∈ S X ) we have Per ( F w ) = P . ◮ We usually just say that F = ( F i ) i is π -independent or w -independent. ◮ Clearly, word independence implies permutation independence; the converse is false. ◮ Questions: Are there word independent SDSs, and is this a common property? Why should we care about this in the first place?

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Properties of π -independent SDS Proposition Let X be a graph and F = ( F i ) i a π -independent sequence of X-local functions with periodic points P. Then each restricted function F i | P : P − → P is a well-defined bijection.

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Properties of π -independent SDS Proposition Let X be a graph and F = ( F i ) i a π -independent sequence of X-local functions with periodic points P. Then each restricted function F i | P : P − → P is a well-defined bijection. Proof. ◮ Let π be a permutation with π (1) = i , let P = Per ( F π ) and P ′ = Per ( F σ 1 ( π ) ) [cyclic 1-shift].

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Properties of π -independent SDS Proposition Let X be a graph and F = ( F i ) i a π -independent sequence of X-local functions with periodic points P. Then each restricted function F i | P : P − → P is a well-defined bijection. Proof. ◮ Let π be a permutation with π (1) = i , let P = Per ( F π ) and P ′ = Per ( F σ 1 ( π ) ) [cyclic 1-shift]. ◮ We have that F i | P : P − → F i ( P ) is a bijection. ◮ From F π (1) ◦ F π = F σ 1 ( π ) ◦ F π (1) it follows that F i ( P ) ⊂ P ′ .

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Properties of π -independent SDS Proposition Let X be a graph and F = ( F i ) i a π -independent sequence of X-local functions with periodic points P. Then each restricted function F i | P : P − → P is a well-defined bijection. Proof. ◮ Let π be a permutation with π (1) = i , let P = Per ( F π ) and P ′ = Per ( F σ 1 ( π ) ) [cyclic 1-shift]. ◮ We have that F i | P : P − → F i ( P ) is a bijection. ◮ From F π (1) ◦ F π = F σ 1 ( π ) ◦ F π (1) it follows that F i ( P ) ⊂ P ′ . ◮ Repeated application of this n times yields | P | = | P ′ | . ◮ Upshot: F i ( P ) = P ′ and by π -independence we have P = P ′ .

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Dynamics Group – A first look ◮ For π -independent SDS each F i | P is a permutation P . ◮ We set F ∗ i := F i | P ◮ If | P | = m and we label the periodic points 1 , 2 , . . . , m , then each F ∗ i ↔ n i ∈ S m .

Introduction Background and Terminology Basic Properties Update Sequence Independent SDS The Dynamics Group Closing Remarks and Outlook Coxeter Groups Dynamics Groups over Circle n Dynamics Group – A first look ◮ For π -independent SDS each F i | P is a permutation P . ◮ We set F ∗ i := F i | P ◮ If | P | = m and we label the periodic points 1 , 2 , . . . , m , then each F ∗ i ↔ n i ∈ S m . Definition (Dynamics group) Let K be a finite set and F = ( F i ) i be a π -independent sequence of X -local functions. The dynamics group of F is G ( F ) = � F ∗ 1 , . . . , F ∗ n � .