Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

Finelli, Michele; Manzini, Giovanni; Margara, Luciano

doi:10.1006/jcom.1998.0474

Cited by 25 publications

(11 citation statements)

References 21 publications

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“…G = Z D ), and X ⊂ A G is a subshift with positive topological entropy, and Φ : A G −→A G is an X-preserving cellular automaton, then the system (X, Φ) is not posexpansive; see [17,Corollary 2] or [18,Theorem 1.1]. The special case when G = Z D and X = A Z D was later reproved in [19,Theorem 4.4]. In this section, we will generalize this result to any symbolic dynamical system satisfying some mild symmetry and mixing conditions.…”

Section: Positive Expansion Versus Network Connectivitymentioning

confidence: 95%

Positive expansiveness versus network dimension in symbolic dynamical systems

Pivato

2011

Theoretical Computer Science

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a b s t r a c t A symbolic dynamical system is a continuous transformation Φ : X −→ X of closed subset X ⊆ A V , where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph ('network') structure on V, whose geometry reveals information about the dynamical system (X, Φ). The dimension dim(V) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim(V) > 1, and the system (A V , X, Φ) satisfies minimal symmetry and mixing conditions, then (X, Φ) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Hölder-continuous.Let X be Cantor space (the compact, perfect, zero-dimensional metrizable topological space, which is unique up to homeomorphism). A Cantor dynamical system is a continuous self-map Φ : X−→X. In addition to its intrinsic interest, the class of Cantor systems is important because it has two universal properties. First, any topological dynamical system on a compact metric space is a factor of a Cantor system; see [1, Corollary 3.9, p.106] or [2, p.1241]. Second, the Jewett-Krieger Theorem says that any ergodic measure-preserving system can be represented as a uniquely ergodic, minimal Cantor system [3, Section 4.4, p.188].If A is a finite set, and V is a countably infinite set, then the product space A V is a Cantor space. Thus, any Cantor dynamical system can be represented as a self-map Φ : A V −→A V , or more generally, as a self-map Φ : X−→X, where X ⊂ A V is a pattern space (a closed subset of A V ). We refer to the structure (A V , X, Φ) as a symbolic dynamical system. At an abstract topological level, any perfect pattern space X is homoeomorphic to Cantor space, so a perfect symbolic dynamical system is simply a Cantor dynamical system. What distinguishes symbolic dynamical systems is a particular way of representing X as a subset of some Cartesian product A V (so that an element of X corresponds to some V-indexed 'pattern' of 'symbols' in the alphabet A).The network of Φ is the digraph structure ( • →) on V defined as follows: for all v, w ∈ V, we have v • → w if and only if the value of Φ(x) w depends nontrivially on the value of x v . We say that (V, • →) has dimension δ if the cardinality of a ball of radius r grows like r δ as r→∞. (Note that δ is not necessarily an integer.) For example, if Φ : A Z D −→A Z D is a cellular automaton, then its network is just a Cayley digraph on Z D ; the dimension of this network is D.This paper explores the relationship between network dimension and the properties of (X, Φ) as a topological dynamical system. In Section 1, we formally define the dimension of a network (V, •→) and the entropy of a pattern space on ...

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Section: Positive Expansion Versus Network Connectivitymentioning

confidence: 95%

Positive expansiveness versus network dimension in symbolic dynamical systems

Pivato

2011

Theoretical Computer Science

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“…For linear CA dynamics reduces to P F(c) (z) = P c (z)A f (z). Manzini et al published a series of articles [4][5][6][7] on topological dynamics of linear 1D CA and explicitly calculated the entropy.…”

Section: Dynamics Of Cellular Automatamentioning

confidence: 99%

“…A new complexity measure is proposed which captures the intrinsic structural complexity of a partitioning of the configuration space, not its randomness. 4. The above concepts are applied to quantify dynamics of generalized graph automata whereas literature is mainly focused on dynamics of cellular automata.…”

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confidence: 99%

Dynamics and Complexity of Computrons

Erkurt

2020

Entropy

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We investigate chaoticity and complexity of a binary general network automata of finite size with external input which we call a computron. As a generalization of cellular automata, computrons can have non-uniform cell rules, non-regular cell connectivity and an external input. We show that any finite-state machine can be represented as a computron and develop two novel set-theoretic concepts: (i) diversity space as a metric space that captures similarity of configurations on a given graph and (ii) basin complexity as a measure of complexity of partitions of the diversity space. We use these concepts to quantify chaoticity of computrons’ dynamics and the complexity of their basins of attraction. The theory is then extended into probabilistic machines where we define fuzzy basin partitioning of recurrent classes and introduce the concept of ergodic decomposition. A case study on 1D cyclic computron is provided with both deterministic and probabilistic versions.

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“…The Lyapunov exponents of cellular automata have been investigated and computed in several papers (see for example [4], [5]) and they are presently a subject of an active research in several examples. The paper [3] contains so called directional Lyapunov exponents which are generalizations of the Shereshevsky concept to space-time semi-group actions.…”

Section: Introductionmentioning

confidence: 99%