Measure rigidity for algebraic bipermutative cellular automata
Mathieu Sablik
Abstract
Let (,F) be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the ×-action of F and the shift map to be the Haar measure on , a closed shift-invariant subgroup of the Abelian compact group . This generalizes simultaneously results of B. Host, A. Maass and S. Mart\'ınez Host-Maass-Martinez-2003 and M. Pivato Pivato-2003. This result is applied to give conditions which also force a (F,)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible expansive cellular automaton on .
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