159 research outputs found
Topological Conjugacies Between Cellular Automata
We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate.
Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant
On some one-sided dynamics of cellular automata
A dynamical system consists of a space of all possible world states and a transformation of said space. Cellular automata are dynamical systems where the space is a set of one- or two-way infinite symbol sequences and the transformation is defined by a homogenous local rule. In the setting of cellular automata, the geometry of the underlying space allows one to define one-sided variants of some dynamical properties; this thesis considers some such one-sided dynamics of cellular automata.
One main topic are the dynamical concepts of expansivity and that of pseudo-orbit tracing property. Expansivity is a strong form of sensitivity to the initial conditions while pseudo-orbit tracing property is a type of approximability. For cellular automata we define one-sided variants of both of these concepts. We give some examples of cellular automata with these properties and prove, for example, that right-expansive cellular automata are chain-mixing. We also show that left-sided pseudo-orbit tracing property together with right-sided expansivity imply that a cellular automaton has the pseudo-orbit tracing property.
Another main topic is conjugacy. Two dynamical systems are conjugate if, in a dynamical sense, they are the same system. We show that for one-sided cellular automata conjugacy is undecidable. In fact the result is stronger and shows that the relations of being a factor or a susbsystem are undecidable, too
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
Infinite sequence of fixed point free pseudo-Anosov homeomorphisms
We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed
points and leaving invariant a sequence of orientable measured foliations on
the same topological surface and the same stratum of the space of abelian
differentials. The existence of such sequences show that all pseudo-Anosov
homeomorphisms fixing orientable measured foliations cannot be obtained by the
Rauzy-Veech induction strategy
Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets
Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them
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