规则26：基本细胞自动机的丰富动力学与符号表示

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本文深入探讨了基本细胞自动机规则26（Elementary Cellular Automaton Rule 26，简称ECA Rule 26）中的复杂动力学。ECA是离散数学领域的一个重要模型，用于模拟简单的物理过程和计算行为。规则26因其丰富的动态特性而备受关注。文章首先从两个核心角度分析：间隔图和子系统。间隔图是一种描绘细胞自动机状态变化的方法，通过观察相邻状态之间的间隔分布，可以揭示自动机的行为模式。在规则26的间隔图中，作者发现了一些关键特征，这表明该自动机在局部规则下展现出了非平凡的动力学结构。接着，文章重点转向了子系统的动力学研究。子系统是指从原始自动机中选取部分状态进行简化后的模型。作者证明了规则26在子系统上具有正的拓扑熵，这意味着它的动力学行为在某种程度上是不可预测的，且随着时间的推移，状态空间的复杂性呈现出增长趋势，这是拓扑可传递性的标志。拓扑熵是衡量动力系统混沌程度的重要指标，正的拓扑熵意味着规则26在子系统上的动力学表现出了高度的复杂性和不可预测性。这种性质通常与分形和混沌现象相联系，暗示着规则26可能存在多种吸引子，即不同的状态序列可能在长时间内保持稳定，但又无法完全确定未来的状态。在符号动力学的框架下，作者进一步探讨了规则26的位移表示。符号动力学是数学分析的一种方法，它通过将状态转换为符号序列来研究系统的长期行为。作者揭示了规则26可以用符号空间上的两个符号移位来刻画，这进一步证实了规则26的动态行为的多样性和灵活性。文章通过细致的分析和研究，揭示了规则26的基本细胞自动机中的复杂动力学特性，包括间隔图的结构、子系统的拓扑性质以及位移表示的符号动力学表达。这些发现不仅加深了对单一规则的理解，也为探索其他细胞自动机规则的动力学行为提供了新的视角。规则26的丰富动力学行为使得它在计算机科学、物理模拟、复杂系统理论等领域具有潜在的应用价值。

COMPLEX DYNAMICS IN ELEMENTARY CELLULAR AUTOMATON 139

termed cells, and x

is the state of cell i. The left shift map (simply shift map)

σ : S

→ S

is deﬁned by [σ (x)]

:= x

i+1

, the right shift map σ

: S

→

is deﬁned by [σ

(x)]

:= x

i−1

for any i ∈ Z. The full shift is the shift

over the alphabet S ={0, 1} on the state space S

. An elementary cellular

automaton (ECA) is a continuous map f

∞

: S

→ S

that is deﬁned by a

local update rule f

: S

→ S as follows: For any conﬁguration x ∈ S

and

cell i ∈ Z,[F

(x)]

= f

[i−1,i+1]

), where “N” stands for the rule number.

For elementary cellular automata (ECA), conﬁgurations are often consid-

ered as inﬁnite in one direction or both directions. Directed graph theory

provides an efﬁcient way to study such cases. A graph G consists of a ﬁnite

set V = V(G) of vertices(or states) together with a ﬁnite set E = E (G)of

edges. Each edge e ∈ E(G) starts at a vertex denoted by i(e) ∈ V(G) and

terminates at a vertex t(e) ∈ V(G), namely, the edge e has initial state i(e)

and terminal state t(e). Sometimes we shorten V(G)toV and E(G)toE

when G is understood. A path π = e

···e

on a graph G is a ﬁnite

sequence of edges e

from G such that t(e

) = i(e

i+1

) for 1 ≤ i ≤ m − 1.

The length of π = e

···e

is |π |=m. The path π = e

···e

starts at

vertex i(π ) = i(e

) and terminates at vertex t(π ) = t(e

), and π is a path

from i(π )tot(π ). A cycle is a path that starts and terminates at the same

vertex. A graph G is irreducible or strongly connected if for every ordered

pair of vertices I and J there is a path in G starting at I and terminating at

J. Let A

denote the number of edges in G with initial state I and terminal

state J. Then the adjacency matrix of G is A = [A

], and its formation from

G is denoted by A = A(G)orA = A

. The edge shift X

or X

is the shift

space over the alphabet S = E speciﬁed by X

= X

={e = (e

)

i∈Z

∈ E

t(e

) = i(e

i+1

) for all i ∈ Z}.

3 INTERVAL MAP OF RULE 26

In this section, we are dedicated to the study of rule 26 via the interval map.

For the conﬁguration ξ

···, its decimal representation is given by

φ =



∞

i=0

−(i+1)

. Let χ

be the interval map induced by F

. Then the

following diagram commutes:

(S)

−−−−→ (S)

⏐



⏐



[0, 1]

−−−−→ [0, 1]

Let I be the interval [0, 1]. Denote by I

the interval [0,

], I

the interval

[

, 1]. For the Bernoulli shift g(φ) = 2φ mod 1, its inverse branches are

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