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Coupled map lattices with asynchronous updatings. (English) Zbl 0981.37016
Consider a family of updating-maps on $$S^{\mathbb{Z}^d}$$, where $$S$$ is the one-dimensional sphere. The Poisson process with values in $$\mathbb{N}^\Lambda$$ is here the counting process. It is induced by the so called maximal causal path $$Q = q_0,\dots,q_n,\dots$$. The system with independent identically Poisson distributed times for individual updatings is studied. Among others the Markov kernels operators, defined as extensions of kernels acting on the product spaces, and the transfer operators for the Markov kernels for classes of updating functions are examined.
The existence and uniqueness of suitable probability measure as well as the exponential decay of correlations are proved.
The details are too complicate to be presented here.

##### MSC:
 37H99 Random dynamical systems 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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