| In the dynamical systems theory, the fundamental objective is to characterize the limit sets of a continuous or differentiable map. For finite-dimensional space, the asymptotic properties do not depend on the metric used. However, in infinite-dimensional space, because of the non-equivalence of the metrics, we can have metric-dependent asymptotic properties for the same map. Beside this, nonlinearity is a necessary condition for chaoticity in finite-dimensional space. But from [1] we can see that this is ho longer true for maps in infinite-dimensional spaces, indeed, a lot of properties of the chaotic linear map are obtained througii the study of the shift map on the l-dimensional lattices function space.In recent years, as the study of cellular neural networks model going further and the need of discretization of various kinds of differential equation , the research on lattices dynamic system has become an increasingly interesting object so that more and more scientific workers pay close attention to. Especially, we can treat the equilibrium solution of an autonomous lattices dynamic as a function defined on all the lattices. At the same time we can see that it is necessary to generalize the lattices function space from l-dimensional to poly-dimensional. In this paper we aim at extending the results of [1] to the function space defined on plane lattices(the functions with range in a finite-dimentional linear normed space), and showing severl dynamic properties for the shift map such as the density of periodic points the density of finite stable set (or called the pre-periodic points set) ?topo-logical transitivity and topological mixing. Moreover, a mistake in [l,Th6 and Th9] is discovered and its revision is given.In order to clarify the conditions for chaoticity and the properties of a linear shift map in the plane lattices function space Fr(N2,X) which is defined asand the shif is defined as (f){m, n) = f(rn + k,n + l),we need first to study the properties of this space. The followings are two of them:With the supremum norm is a non-separable Banach space. If compact in compact, then F1{N2,Bm,n) is compact in Fr(N2,X).Secondly, we proved that,for all k,l N,the shift map : Fr(N2,X) Fr(N2,X) is (1) surjetive; (2) linear; (3) continuous; (4) non-compact and (5) not invert-ible. Furthermore, its norm is r-k1r2-l The point-wise spectrum of which is defined in The eigenspace associated with each eigenvalue is The linear subspace of Fr(N2,C) generated by , t are respectively the k, l-roots of unity.} is a proper subset of periodic points of in Fr(N2, C). Especially, The linear subspace of Fr(N2,C) generated by Kp,q is a proper subset of fixed points of in Fr(N2,C). So we see there is a mistake in th6[l].Then the following criteria are established as the state space is compact.Let B be a. compact subset of X with more than one point. Then k > 0, l > 0, k + l > 0, is chaotic in Fr(N2, 5).Restricted on and T2 = C1 x C1.the shift map has sensitivity to initial conditions and density of periodic points, but it is not transitive.The complex behavior of the shift map in the whole space Fr(N2,X) can be further underlined by the analysis of the stable sets. Let be a map in a metric space A'. For any x X,we denote the stable set Sx of x as .The finite stable set S*x( Sx) of x is defined as Sx* = .Let be a shift map in The conclusions above show the dynamic properties of the shift map in infinite-dimension space Fy(N2,X), (0 < r1< 1,0 < r2 < l),and provide us another example of chaotic linear map.Finally, we extend the analysis of the shift map to suitable Banach space of bi-infinite plane lattices function space. The main difference with respect to the previous case is that the shift map is now invertible, and the inverse (f)(m,n) = f(m - k, n - I) has the same properties of . It is easy to show that all the notations and conclusions are almost the same. What we are interested in is the chaoticity of the shift map is chaotic and topologically mixing in Fr(Z2, B), B compact in X. (2)...
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