2002/08/01 by MARCUS PIVATO, REEM YASSAWI
paper · doi:10.1017/s0143385702000548
Let \mathbbM be a monoid (e.g. ℕ , ℤ , or \mathbbMD ), and A an abelian group. A^\mathbbM is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrakF:A^\mathbbM\longrightarrowA^\mathbbM that commutes with all shift maps. Let μ be a (possibly non-stationary) probability measure on A^\mathbbM ; we develop sufficient conditions on μ and \mathfrakF so that the sequence \\mathfrakFNμ\N=1^∞ weak* converges to the Haar measure on A^\mathbbM in density (and thus, in Cesàro average as well). As an application, we show that, if A=ℤ/p ( p prime), \mathfrakF is any ‘non-trivial’ LCA on A(ℤD) , and μ belongs to a broad class of measures (including most Bernoulli measures (for D ≥ 1 ) and ‘fully supported’ N -step Markov measures (when D=1 )), then \mathfrakFNμ weak* converges to the Haar measure in density.