On the Interplay of Direct Topological Factorizations and Cellular Automata Dynamics on Beta-Shifts

Abstract

<p>We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts <em>S_β</em> and its relation to direct topological factorizations. We show that any reversible CA <em>F:S_β→S_β</em> has an almost equicontinuous direction whenever <em>S_β</em> is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations <em>X×Y</em> of two nontrivial subshifts <em>X</em> and <em>Y</em>. We also give a simple criterion to determine whether <em>S_nγ</em> is conjugate to <em>S_n×S_γ</em> for a given integer <em>n</em>≥1 and a given real <em>γ</em>>1 when <em>S_γ</em> is a subshift of finite type. When <em>S_γ</em> is strictly sofic, we show that such a conjugacy is not possible at least when <em>γ</em> is a quadratic Pisot number of degree 2. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.<br></p>