Julien Cervelle, Enrico Formenti, Pierre Guillon
The trace subshift of a cellular automaton is the subshift of all possible columns that may appear in a space-time diagram, ie the infinite sequence of states of a particular cell of a configuration; in the language of symbolic dynamics one says that it is a factor system. In this paper we study conditions for a sofic subshift to be the trace of a cellular automaton.
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Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is then a compact abelian group; a `linear cellular automaton' (LCA) is a topological group endomorphism Φ:A^M --> A^M that commutes with all shift maps. Suppose μis a probability measure on A^M whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φand μ) under which Φ`asymptotically randomizes' μ, meaning that wk*lim_{J\ni j --> oo} Φ^j μ= η, where ηis the Haar measure on A^M, and J has Cesaro density 1. In the case when Φ=1+σ, we provide a condition on μthat is both necessary a…
We study limit sets of stable cellular automata standing from a symbolic dynamics point of view where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto…
Boolean TQFTs with accumulating defects, sofic systems, and automata for infinite words
本文扩展了有限状态自动机与布尔一维TQFT的对应关系，将其推广到缺陷累积情形，并关联sofic系统与$\omega$-自动机。
Generalized Gottschalk's conjecture for sofic groups and applications
该研究将元胞自动机的满射性与对偶满射性定理推广到sofic群上元胞自动机的局部及全局一致有界奇异扰动情形，得到代数非均匀元胞自动机的满射性，并证明sofic群上扭群环的稳定有限性，推广了Kaplansky相关猜想的已知结果。
We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts $S_β$. We show that any reversible CA $F:S_β\to S_β$ has an almost equicontinuous direction whenever $S_β$ is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime. We also make some preliminary observations on direct topological factorizations of beta-shifts that are subshifts of finite type.
We prove that topologically isomorphic linear cellular automaton shifts are algebraically isomorphic. Using this, we show that two distinct such shifts cannot be isomorphic. We conclude that the automorphism group of a linear cellular automaton shift is a finitely generated abelian group.
The Curious Case of Reversible Elementary Second Order Cellular Automaton 115
证明了可逆基本二阶元胞自动机规则115在有限初始配置下是周期的，并研究了具有有趣周期函数的有限配置族。
Physical universality of a cellular automaton was defined by Janzing in 2010 as the ability to implement an arbitrary transformation of spatial patterns. In 2014, Schaeffer gave a construction of a two-dimensional physically universal cellular automaton. We construct a one-dimensional version of the automaton.
We are interested in topological and ergodic properties of one dimensional cellular automata. We show that an ergodic cellular automaton cannot have irrational eigenvalues. We show that any cellular automaton with an equicontinuous factor has also as a factor an equicontinuous cellular automaton. We show also that a cellular automaton with almost equicontinuous points according to Gilman's classification has an equicontinuous measurable factor which is a cellular automaton. 2000 Mathematics Subject Classification.: 37B15, 54H20, 37A30. Key words and phrases. Cellular Automata, Dynamical system…
We construct a one-dimensional uniquely ergodic cellular automaton which is not nilpotent. This automaton can perform asymptotically infinitely sparse computation, which nevertheless never disappears completely. The construction builds on the self-simulating automaton of Gács. We also prove related results of dynamical and computational nature, including the undecidability of unique ergodicity, and the undecidability of nilpotency in uniquely ergodic cellular automata.
Dynamical Properties of A Cellular Automaton Model of Language Shift in Algeria
本文研究了模拟阿尔及利亚语言更替的二维元胞自动机的动力学性质，探讨了其满射性与等度连续点，并通过计算机模拟与解析证明获得了不动点吸引域的性质。
The asymptotic behavior of a cellular automaton iterated on a random configuration is well described by its limit probability measure(s). In this paper, we characterize measures and sets of measures that can be reached as limit points after iterating a cellular automaton on a simple initial measure, in the same spirit as SRB measures. In addition to classical topological constraints, we exhibit necessary computational obstructions. With an additional hypothesis of connectivity, we show these computability conditions are sufficient by constructing a cellular automaton realising these sets, usin…
We present a new sufficient criterion to prove that a non-sofic half-synchronized subshift is direct prime. The criterion is based on conjugacy invariant properties of Fischer graphs of half-synchronized shifts. We use this criterion to show as a new result that all n-Dyck shifts are direct prime, and we also give new proofs of direct primeness of non-sofic beta-shifts and non-sofic S-gap shifts. We also construct a class of non-sofic synchronized direct prime subshifts which additionally admit reversible cellular automata with all directions sensitive.
In this paper, we give a singular function on a unit interval derived from the dynamic of the one-dimensional elementary cellular automaton Rule 150. We describe properties of the resulting function, that is strictly increasing, uniformly continuous, and differentiable almost everywhere, and we show that it is not differentiable at dyadic rational points. We also give functional equations that the function satisfies, and show that the function is the only solution of the functional ones.
No weakly factor-universal cellular automaton
本文研究了是否存在一个细胞自动机，使得所有细胞自动机都是其因子。作者证明不存在这样的自动机，并探讨了弱因子与周期点周期的关系。
A perturbed cellular automaton with two phase transitions for the ergodicity
本文利用Gács结果构造了一个具有两种遍历性相变的元胞自动机，证明其在小噪声速率下遍历、中等速率下非遍历、接近1的速率下再次遍历。
Using an unusual, yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov Exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable entropy. The measure m we consider gives the m-expansiveness property to the automaton. It is constructed with respect to a factor dynamical system based on simple "counter dynamics". As a counterpart, we …
Limit dynamics of elementary cellular automaton 18
本文通过极限集、泛型极限集和$μ$-极限集研究初等元胞自动机18的渐近动力学，利用kinks局部模式刻画泛型极限集配置，并证明其三个极限集互不相同。
We address the dynamics of the cellular automaton (CA) that multiplies by $p/q$ in base $pq$ (for coprime $p>q>1$) by studying its trace subshift. We present a conjugacy of the trace to a previously studied base-$p/q$ numeration system. We also show that the trace subshift is not synchronizing and in particular not sofic. As a byproduct we compute its complexity function and we conclude by presenting an example of a sofic shift with the same complexity function.
The total activity of the single-seeded cellular rule 150 automaton does not follow a one-step iteration like other elementary cellular automata, but can be solved as a two-step vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows to compute the total activity time series more efficiently than by simulating the whole spatio-temporal process, or even by using the closed expression.
We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Törmä that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage …
We show that an almost expansive cellular automaton according to Gilman's classification cannot have irrational eigenvalues. This completes the proof that any surjective cellular automaton cannot have irrational eigenvalues for the uniform measure.
In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular autom…
Let $G$ be a group. Let $X$ be a connected algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of $K$-points of $X$. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over $(G,X,K)$. They are cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M\subset G$ is a finite memory set of $τ$. Our first result is that when $G$ is sofic, such an algebraic group cellular automaton $τ$ is invertible whenever it is injective and $\text{char}(K)=0…
We study one-dimensional cellular automata evolutions with both temporal and spatial periodicity. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period $σ$ and number of states $n$. When $σ= 2, 3, 4$, or $6$, and we restrict the rules to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. We also construct non-additive rules with temporal period on the same order as the trivial upper bound $n^σ$. Experimental results, open problems, and possible extensions of o…
In this article we give a new definition of some analog of Lyapunov exponents for cellular automata . Then for a shift ergodic and cellular automaton invariant probability measure we establish an inequality between the entropy of the automaton, the entropy of the shift and the Lyapunov exponent.
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