On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime p

Abstract

A method is known by which any integer \(\, n\geq2\,\) in a metric Cantor space of right-infinite words \(\,\tilde{A}_{n}^{\,\mathbb N}\,\) gives a construction of a non-injective cellular automaton \(\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,\) which is chaotic in Devaney sense, has a radius \(\, r=1,\,\) continuum of fixed points and topological entropy \(\, log(n).\,\) As a generalization of this method we present for any integer \(\, n\geq2,\,\) a construction of a cellular automaton \(\,(A_{n}^{\,\mathbb{N}},\, F_{n}),\,\) which has the listed properties of \(\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,\) but has no fixed points and has continuum of periodic points with the period 2. The construction is based on properties of cellular automaton introduced here \(\,(B^{\,\mathbb N},\, F)\,\) with radius \(1\) defined for any prime number \(\, p.\,\) We prove that \(\,(B^{\,\mathbb N},\, F)\,\) is non-injective, chaotic in Devaney sense, has no fixed points, has continuum of periodic points with the period \(2\) and topological entropy \(\, log(p).\,\)

Keywords

• one-sided cellular automata
• D-chaotic
• E-chaotic
• fixed points
• topological entropy