平面网格函数空间上的线性混沌映射
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摘要
动力系统主要研究连续或可微映射的极限集性质，有限维空间上的序列的收敛性与度量无关，而无限维空间上由于度量间的不等价性，对同一映射会有依赖于不同的度量的不同的渐近性质。另外在有限维空间中非线性是混沌的必要条件，而从文献[1]可看到这在无限维空间中是不成立的，事实上，文献[1]通过定义在一维整数格点上的函数空间中的移位映射的研究，得出了许多关于混沌线性映射的性质。
近年来，随着神经网络模型研究的深入和各类微分方程离散化研究的需要，格点动力系统的研究方兴未艾，特别的，可将一个自治的格点动力系统的平衡解视为一个定义在全体格点上的函数，同时显见有必要将文献[1]中的一维格点函数空间推广到多维格点函数空间，本文主要是将文献[1]的结果推广到平面网格函数空间(函数取值于有限维赋范空间)上，并研究移位映射的周期点集的稠密性、有限稳定集(亦称准周期点集)的稠密性、拓扑传递性和拓扑混合性等动力学性态，我们在文中还指出并修正了文献[1]中定理6中的错误，并纠正[1]定理9得不正确提法。
本文研究平面网格函数空间F_γ(N~2，X)上的线性映射σ_1~kσ_2~l产生混沌的条件及其性质，其中
F_γ(N~2，X)={f∣f：N~2→X，(?){∣f(m，n)∣γ_1~mγ_2~n＜+∞}}，σ_1~kσ_2~l(f)(m，n)=f(m+k，n+l)为网格函数空间上的移位映射。
首先本文定义了空间F_γ(N~2，X)，并研究其基本性质。
F_γ(N~2，X)关于范数∣·∣_γ为一不可分Banach空间，设0＜γ=(γ_1，γ_2)＜1，和B_(m，n)，B均为X中紧集，且B_(m，n)(?)B，(?)_(m，n)∈N，则F_γ(N~2，B_(m，n))为F_γ(N~2，X)中的紧集。
其次本文主要介绍F_γ(N~2，X)上的移位映射σ_1~kσ_2~l，并研究它们的性质，得出：
对任意k，l∈N，移位映射σ_1~kσ_2~l：F_γ(N~2，X)→F_γ(N~2，X)是(1)满的，(2)线性的，(3)连续的，(4)非紧的，(5)非可逆的，且σ_1~kσ_2~l的范数为γ_1~(-k)γ_2~(-l)。
σ_1~kσ_2~l的点态潜可表为λ_1~kλ_2~l，其中0≤λ_1≤γ_1~(-1)，0≤λ_2≤γ_2~(-1)，其相应的特征子空间为{f∈F_γ(N~2，X)∣f(m，n)=λ_1~(m-k)λ_2~(n-l)x，(?)_(m，n)∈N，x∈X}，
由K_(k，l)={(μ_s，(?)_t)~((?))，s=0，1，2，…，k-1，t=0，1，2，…，l-1，μ_s，(?)_t分别为单位元的k，l次单位根}生成的线性子空间属于σ_1~pσ_2~q在F_γ(N~2，C)中的周期点集，特别，由K_(p，q)生成的线性子空间属于其不动点集，从此结论可知文献[1]定理6的结论有误。
再次本文主要研究当状态空间为紧致时，移位映射的σ_1~kσ_2~l的各种混沌性质，得出：
设B为X中非空紧子集，B含X中不止一点，则(?)_k≥0，l≥0，k+l＞0，σ_1~kσ_2~l在F_γ(N~2，B)中混沌且拓扑混合。
移位映射。全码限制在(紧)不变子集H二CIK和HI二TZK_l:，保留了刘初始
条件的敏感性和周川1点稠密性但不满足拓扑传递性.
其中兀={(。2“0，。27T，T)(，):0，二〔!o，1!}，Cl={e〔C}}〔:}=1}，TZ=
CI x Cl.
我们通过刘稳定集的进一步分析来研究移位映射a全a玉在空间尽(NZ，X)上的
复杂行为.设厂是度量空间X_I:-的映射.对任意x〔X，记x关于f的稳定集为又=
{川lim。一oofm(y)二对，二关于f的有限稳定集为义二{:旧二:任N，〕广”(:)=
对c Sx.时码为凡(妒，x)一L的移位映射，对任意f〔尽(NZ，X)，f关于。全a毛的
有限稳定集SJ的闭包包含Fl(N，，X).
上述儿个结论揭示了移位映射在无限维空间凡(NZ，X)，(守1<1，守: 动力系统性质，提供了一个混沌线性映射的例子.
最后本文将以__l:分析延伸到适当的双向无限平面网格函数空间.与前面讨论的
移位映射主要区别在一」几这里的移位映射是可逆的，定义移位映射a全a吕的逆映射为
二，一”。牙‘(.厂)(，，，，，‘卜厂(，，卜无，，卜l).
且J一1与J有相同的性质.容易证明所有的定义及符号及定理结论基本与前面相同.我
们最感兴趣的是a全码的混沌性:
(1)时码在凡(22，B”中混沌且拓扑混合，其中.B为X中的紧集.(2)移位映射
砖码限制在(紧)不变子集H_L，保留了刘初始条件的敏感性和周期点稠密性但不满
足拓扑传递性.
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 as
and 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)
近年来，随着神经网络模型研究的深入和各类微分方程离散化研究的需要，格点动力系统的研究方兴未艾，特别的，可将一个自治的格点动力系统的平衡解视为一个定义在全体格点上的函数，同时显见有必要将文献[1]中的一维格点函数空间推广到多维格点函数空间，本文主要是将文献[1]的结果推广到平面网格函数空间(函数取值于有限维赋范空间)上，并研究移位映射的周期点集的稠密性、有限稳定集(亦称准周期点集)的稠密性、拓扑传递性和拓扑混合性等动力学性态，我们在文中还指出并修正了文献[1]中定理6中的错误，并纠正[1]定理9得不正确提法。
本文研究平面网格函数空间F_γ(N~2，X)上的线性映射σ_1~kσ_2~l产生混沌的条件及其性质，其中
F_γ(N~2，X)={f∣f：N~2→X，(?){∣f(m，n)∣γ_1~mγ_2~n＜+∞}}，σ_1~kσ_2~l(f)(m，n)=f(m+k，n+l)为网格函数空间上的移位映射。
首先本文定义了空间F_γ(N~2，X)，并研究其基本性质。
F_γ(N~2，X)关于范数∣·∣_γ为一不可分Banach空间，设0＜γ=(γ_1，γ_2)＜1，和B_(m，n)，B均为X中紧集，且B_(m，n)(?)B，(?)_(m，n)∈N，则F_γ(N~2，B_(m，n))为F_γ(N~2，X)中的紧集。
其次本文主要介绍F_γ(N~2，X)上的移位映射σ_1~kσ_2~l，并研究它们的性质，得出：
对任意k，l∈N，移位映射σ_1~kσ_2~l：F_γ(N~2，X)→F_γ(N~2，X)是(1)满的，(2)线性的，(3)连续的，(4)非紧的，(5)非可逆的，且σ_1~kσ_2~l的范数为γ_1~(-k)γ_2~(-l)。
σ_1~kσ_2~l的点态潜可表为λ_1~kλ_2~l，其中0≤λ_1≤γ_1~(-1)，0≤λ_2≤γ_2~(-1)，其相应的特征子空间为{f∈F_γ(N~2，X)∣f(m，n)=λ_1~(m-k)λ_2~(n-l)x，(?)_(m，n)∈N，x∈X}，
由K_(k，l)={(μ_s，(?)_t)~((?))，s=0，1，2，…，k-1，t=0，1，2，…，l-1，μ_s，(?)_t分别为单位元的k，l次单位根}生成的线性子空间属于σ_1~pσ_2~q在F_γ(N~2，C)中的周期点集，特别，由K_(p，q)生成的线性子空间属于其不动点集，从此结论可知文献[1]定理6的结论有误。
再次本文主要研究当状态空间为紧致时，移位映射的σ_1~kσ_2~l的各种混沌性质，得出：
设B为X中非空紧子集，B含X中不止一点，则(?)_k≥0，l≥0，k+l＞0，σ_1~kσ_2~l在F_γ(N~2，B)中混沌且拓扑混合。
移位映射。全码限制在(紧)不变子集H二CIK和HI二TZK_l:，保留了刘初始
条件的敏感性和周川1点稠密性但不满足拓扑传递性.
其中兀={(。2“0，。27T，T)(，):0，二〔!o，1!}，Cl={e〔C}}〔:}=1}，TZ=
CI x Cl.
我们通过刘稳定集的进一步分析来研究移位映射a全a玉在空间尽(NZ，X)上的
复杂行为.设厂是度量空间X_I:-的映射.对任意x〔X，记x关于f的稳定集为又=
{川lim。一oofm(y)二对，二关于f的有限稳定集为义二{:旧二:任N，〕广”(:)=
对c Sx.时码为凡(妒，x)一L的移位映射，对任意f〔尽(NZ，X)，f关于。全a毛的
有限稳定集SJ的闭包包含Fl(N，，X).
上述儿个结论揭示了移位映射在无限维空间凡(NZ，X)，(守1<1，守:
最后本文将以__l:分析延伸到适当的双向无限平面网格函数空间.与前面讨论的
移位映射主要区别在一」几这里的移位映射是可逆的，定义移位映射a全a吕的逆映射为
二，一”。牙‘(.厂)(，，，，，‘卜厂(，，卜无，，卜l).
且J一1与J有相同的性质.容易证明所有的定义及符号及定理结论基本与前面相同.我
们最感兴趣的是a全码的混沌性:
(1)时码在凡(22，B”中混沌且拓扑混合，其中.B为X中的紧集.(2)移位映射
砖码限制在(紧)不变子集H_L，保留了刘初始条件的敏感性和周期点稠密性但不满
足拓扑传递性.
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 as
and 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)
引文
[1] Lenci S, Lupini R. Chaotic linear maps, Nonlinear Analysis, TMA, 2001, 45: 707-721.
[2] Banks J, Brooks J. Cairns G, et al. On Devaney's definition of chaos. Amer.Math. Monthly, 1992, 99: 332-334.
[3] 陈芳跃．CNN符号动力系统．上海大学理学院博士学位论文，2003．10．
[4] Devaney R L. An Introduction to Chaotical Systems. Redwood City: Addison-Wesley Publishing Company, 1987; 2nd edition, 1989.
[5] Martelli M, Dang M, Seph T. Defining Chaos. Math. Magazine, 1998, 71(2): 112-122.
[6] Robinson C. Dynamical Systems: Stability, Symbolic Dynamics and Chaos. Florida: CRC Press, 1995.
[2] Banks J, Brooks J. Cairns G, et al. On Devaney's definition of chaos. Amer.Math. Monthly, 1992, 99: 332-334.
[3] 陈芳跃．CNN符号动力系统．上海大学理学院博士学位论文，2003．10．
[4] Devaney R L. An Introduction to Chaotical Systems. Redwood City: Addison-Wesley Publishing Company, 1987; 2nd edition, 1989.
[5] Martelli M, Dang M, Seph T. Defining Chaos. Math. Magazine, 1998, 71(2): 112-122.
[6] Robinson C. Dynamical Systems: Stability, Symbolic Dynamics and Chaos. Florida: CRC Press, 1995.