期刊名称：International Journal of Networking and Computing
印刷版ISSN：2185-2847
出版年度：2017
卷号：7
期号：2
页码：124-135
语种：English
出版社：International Journal of Networking and Computing
摘要：Linear cellular automata have many invariant measures in general. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure - the most natural one. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. In the two-state case, we characterize when iterates by addition modulo 2 cellular automata starting from a convex combination of strong mixing probability measures can converge. This also gives all invariant measures inside the class of those probability measures. We can obtain a similar result for iterates by addition modulo an odd prime number cellular automata starting from strong mixing probability measures.
其他摘要：Linear cellular automata have many invariant measures in general. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure - the most natural one. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. In the two-state case, we characterize when iterates by addition modulo 2 cellular automata starting from a convex combination of strong mixing probability measures can converge. This also gives all invariant measures inside the class of those probability measures. We can obtain a similar result for iterates by addition modulo an odd prime number cellular automata starting from strong mixing probability measures.
