Uniform Bernoulli Measure Research Articles

Dynamical properties of expansive one-sided cellular automata

To every one-sided one-dimensional cellular automatonF with neighbourhood radiusr we associate its canonical factor defined by considering only the firstr coordinates of all the images of points under the powers ofF. Whenever the cellular automaton is surjective, this factor is a subshift which plays a primary role in its dynamics. In this article we study the class of positively expansive one-sided cellular automata, i.e. those that are conjugate to their canonical factors. This class is a natural generalisation of the toggle or permutative cellular automata introduced in [He]. We prove that the canonical factors of positively expansive one-sided cellular automata are mixing subshifts of finite type that are shift equivalent to full shifts. Moreover, the uniform Bernoulli measure is the unique measure of maximal entropy forF. Consequently, their natural extensions are Bernoulli. We also describe a family of non-permutative positively expansive cellular automata.