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There are three types of distributional chaos, namely DC1, DC2 and DC3. In this paper we present two constant-length substitution systems, one is DC2 but not DC1, and the other is DC3 but not DC2. (In this paper, chaos means existence of an uncountable scrambled set of the corresponding type while the existing examples deal with single pairs of points only.) 相似文献
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Heng Liu Lidong Wang Zhenyan Chu 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6144-6147
Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence. 相似文献
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The aim of this note is to use methods developed by Kuratowski and Mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence. In particular, we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to Li and Yorke. Namely, positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence, which is not the case for distributional chaos as introduced by Schweizer and Smítal. 相似文献
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J. Dvo?áková 《Communications in Nonlinear Science & Numerical Simulation》2012,17(2):785-787
We show that if f is a DC3 continuous map of a compact metric space then also fN is DC3, for every N > 0. This solves a problem given by [Li R. A note on the three versions of distributional chaos. Commun Nonlinear Sci Numer Simulat 2011;16:1993-1997]. 相似文献
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We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces c0 and ?1 has a subspace which has the AP but fails to have the strong AP. 相似文献
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Félix Martínez-Giménez 《Journal of Mathematical Analysis and Applications》2009,351(2):607-1220
We provide sufficient conditions which give uniform distributional chaos for backward shift operators. We also compare distributional chaos with other well-studied notions of chaos for linear operators, like Devaney chaos and hypercyclicity, and show that Devaney chaos implies uniform distributional chaos for weighted backward shifts, but there are examples of backward shifts which are uniformly distributionally chaotic and not hypercyclic. 相似文献
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《Chaos, solitons, and fractals》2007,31(2):347-355
We give sufficient conditions for a shift to be distributionally chaotic and chaotic in the sense of Li and Yorke. In the case of cocyclic shifts we show the equivalence between distributional chaos, chaos in the sense of Li and Yorke, positivity of entropy and uncountability of subshift. 相似文献
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设(X,d)是紧致度量空间.设(K,H)是X中所有非空紧子集所组成的空间,并赋予由d导出的Hausdorff度量H.主要探讨了拓扑动力系统(X,G)的混合性、混沌和集值动力系统(K,G)的混合性、混沌之间的关系,其中G是拓扑群. 相似文献
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Vegard Lima 《Journal of Mathematical Analysis and Applications》2007,334(1):593-603
We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y. 相似文献
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F. Balibrea 《Topology and its Applications》2009,156(9):1673-1678
In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994) 737-854] for continuous maps of the interval. We show that there is a DC1 homeomorphism F∈T such that any ω-limit set contains unique minimal set. This homeomorphism is constructed such that it is increasing on some fibres, and decreasing on the other ones. Consequently, F has zero topological entropy. Similar behavior is impossible when F is nondecreasing on the fibres, as shown by Paganoni and Smítal [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26 (2005) 581-589]. This result contributes to the solution of an old problem of Sharkovsky concerning classification of triangular maps but it is interesting by itself since it implies interesting open problems concerning relations between DC1 and minimality. 相似文献
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The main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic. 相似文献
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Berthold Schweizer 《Milan Journal of Mathematics》1996,66(1):159-167
The purpose of this paper is to indicate how the theory of distributional chaos was motivated by certain constructs from the theory of probabilistic metric spaces, to introduce the notion of distributional chaos and to illustrate some of its features with a simple example. 相似文献
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Let R be a commutative ring with identity. We will say that an R-module M satisfies the weak Nakayama property, if IM=M, where I is an ideal of R, implies that for any x∈M there exists a∈I such that (a−1)x=0. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R is a local ring, then R is a Max ring if and only if J(R), the Jacobson radical of R, is T-nilpotent if and only if every R-module satisfies the weak Nakayama property. 相似文献
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There are three versions of distributional chaos, namely DC1, DC2 and DC3. By using an example of constant-length substitution system, we show that DC3 need not imply Li–Yorke chaos. (In this paper, chaos means the existence of an uncountable scrambled set of the corresponding type, while the existing example only deals with a single pair of points.) 相似文献
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We give a full topological characterization of omega limit sets of continuous maps on graphs and we show that basic sets have similar properties as in the case of the compact interval. We also prove that the presence of distributional chaos, the existence of basic sets, and positive topological entropy (among other properties) are mutually equivalent for continuous graph maps. 相似文献
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Consider a random vector, and assume that a set of its moments information is known. Among all possible distributions obeying the given moments constraints, the envelope of the probability distribution functions is introduced in this paper as distributional robust probability function. We show that such a function is computable in the bi-variate case under some conditions. Connections to the existing results in the literature and its applications in risk management are discussed as well. 相似文献
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Zeng Guangxing 《数学学报(英文版)》1998,14(4):481-486
In this paper, we prove the main result: Let both (K, S) and (K
*,S
*) be preordered fields, and let (K
*,S
*) be a finitely generated extension of (K, S). IfK
* is transcendental overK, then (K
*,S
*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover,
some results on the weak Hilbert property are established.
Project supported by National Natural Science Foundation of China 相似文献