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仿射元胞自动机的极限测度

Limit Measures for Affine Cellular Automata

Marcus Pivato, Reem Yassawi

备注：
LaTeX2E Format, revised version contains minor corrections
发表信息：
Ergodic Theory & Dynamical Systems (August, 2002) v.22 (#4) pp 1269-1287
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摘要：

设M是么半群（例如格Z^D），A是阿贝尔群。A^M则是一个紧阿贝尔群;线性元胞自动机（LCA）是一个连续自同态F：A^M --> A^M，它与所有移位映射交换。设μ是A^M上的一个（可能是非平稳的）概率测度，我们给出了μ和F的充分条件，使得序列{F^N μ}（N= 1，2，3，.）弱 *-收敛到A^M上的Haar测度，在密度上（因此，也在Cesaro平均上）。作为应用，我们证明了：如果A=Z/p（p素数），F是A^{（Z^D）}上的任何"非平凡“LCA，μ属于一个广泛的测度类（包括大多数Bernoulli测度（当D >= 1时）和”完全可测的“N步马尔可夫测度（当D=1时）），则F^N μ弱 *-依密度收敛于Haar测度.

英文摘要：

Let M be a monoid (e.g. the lattice Z^D), and A an abelian group. A^M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. Let mu be a (possibly nonstationary) probability measure on A^M; we develop sufficient conditions on mu and F so that the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density (and thus, in Cesaro average as well). As an application, we show: if A=Z/p (p prime), F is any ``nontrivial'' LCA on A^{(Z^D)}, and mu 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 F^N mu weak*-converges to Haar measure in density.

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