Topological Entropy of Group Cellular Automata is Effectively Computable
Niccolò Castronuovo, Alberto Dennunzio, Jarkko Kari, Luciano Margara
原文摘要
Topological entropy is a fundamental invariant of dynamical systems, quantifying their complexity. For cellular automata, however, determining topological entropy is generally algorithmically intractable. We study group cellular automata, namely cellular automata over a finite group alphabet whose global map is a group endomorphism. In dimension one, we decompose topological entropy into two one-sided components describing information propagating from the left and from the right in space--time. We introduce finite families of left and right traces and prove that the entropy is the sum of their growth rates. Hence, the entropy is completely determined by the growth rate of a fixed-width family of space--time patterns. This reduction from arbitrarily large windows to uniformly bounded width yields an effective procedure for computing the entropy. In particular, the entropy of every one-dimensional group cellular automaton is the logarithm of a positive integer. Using Pontryagin duality and the Bridge Theorem between topological and algebraic entropy, we also derive an explicit formula for linear cellular automata over $(\mathbb{Z}/p\mathbb{Z})^n$ in terms of their characteristic polynomial. Moreover, in every dimension $D\geq 2$, a group cellular automaton is either equicontinuous with zero entropy or sensitive to initial conditions with infinite entropy. Combined with decidability of equicontinuity, this implies that topological entropy is effectively computable from the local rule in every dimension. Thus, group cellular automata form a natural nontrivial class in which algebraic structure restores computability of topological entropy.
cs.CC · math.DS