In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

中文翻译：

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一维格林伯格-黑斯廷斯元胞自动机的动力学和拓扑熵

在本文中，我们分析了可激发媒体的非游走一维 Greenberg-Hastings 元胞自动机模型$e\geqslant 1$兴奋和$r\geqslant 1$耐火状态并确定其（严格正）拓扑熵。我们证明它是由非游走集的 Devaney 混沌封闭不变子集产生的，该集由碰撞和湮灭行波组成，它与耦合移位动力学的斜积动力学系统共轭。此外，我们将非游走集的剩余部分明确地确定为具有严格较少拓扑熵的马尔可夫系统，该系统对于大型$e,r$.