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Spacetime chaos in coupled map lattices. (English) Zbl 0679.58028
The authors construct some examples of infinite-dimensional dynamical systems which are naturally described by two-dimensional lattice models of statistical mechanics. The authors assert that their results give a more or less complete description of spacetime chaos. Let \(M=\prod_{i\in Z}I_ i\) be the space of all doubly infinite sequences \((x_ i)_{i\in Z}\), \(x_ i\in I_ i=[0,1]\) for \(i\in Z\). M has a natural topology and the induced \(\sigma\)-algebra U. Consider a function f: \(I\to R\) of class \(C^{1+\gamma}\) \((\gamma >0)\) such that (1) \(f(0)=0\), \(f(1)=d\geq 2\), where \(d\in Z\) and (2) \(f'\geq \lambda =const>1\). for \(x=(x_ i)\in M\) we denote by \(y_ i\) the fractional part of the real number \(f(x_ i)\) and define the map F: \(M\to M\) by \((Fx)_ i=y_ i\) for \(i\in Z\). Next, we take a function a: \(I\to R\) of class \(C^ 2\) such that (i) \(a(y)=\epsilon\) for \(\delta\leq y\leq 1-\delta\), (ii) \(a(0)=a(1)=0\); (iii) \(a'(y)\geq 0\) for \(0\leq y\leq \delta\), \(a'(y)\leq 0\) for 1-\(\delta\leq y\leq 1\), and put \(a^{(1)}=\max \{| a'(y)|\); \(y\in I\}\). Define the averaging operator A: \(M\to M\) by \((Ax)_ i=(1- a(x_ i))x_ i+a(x_ i)(x_{i-1}+x_{i+1})/2\) for \(x=(x_ i)\in M\) and \(i\in Z\). Put \(\Phi =A\circ F\). The main result of the paper is the following: For sufficiently small \(\epsilon\), \(a^{(1)}\) there exists a measure \(\mu\) on (M,U) such that (1) \(\mu\) is \(\Phi\)-invariant, (2) for any integers \(N_ 1\), \(N_ 2>0\) the induced measure on the space of finite sequences \(\{x_ i\}\), \(-N_ 1\leq i\leq N_ 2\) is absolutely continuous with respect to the Lebesgue measure, (3) the dynamical systems (M,U,\(\mu\),\(\Phi)\) and (M,U,\(\mu\),S) are both mixing, where S is the shift in M.
Reviewer: A.Morimoto

MSC:
37A99 Ergodic theory
28D99 Measure-theoretic ergodic theory
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