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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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