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1.
非自治动力系统的原像熵 总被引:4,自引:0,他引:4
本文对紧致度量空间上的连续自映射序列应用生成集和分离集引入了点原像熵、原像分枝熵以及原像关系熵等几类原像熵的定义并进行了研究.主要结果是:(1) 证明了这些熵都是等度拓扑共轭不变量.(2)讨论了这些原像熵之间及它们与拓扑熵之间的关系,得到了联系这些熵的不等式.(3)证明了对正向可扩的连续自映射序列而言, 两类点原像熵相等,原像分枝熵与原像关系熵也相等.(4)证明了对(a).由闭Riemann 流形上的一个扩张映射经充分小的C1-扰动生成的自映射序列,以及(b).有限图上等度连续的自映射序列,有零原像分枝熵. 相似文献
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类似于拓扑熵,点态原像熵作为动力系统的不变量,也度量了紧度量空间上系统的复杂性.但至今不知其性质与拓扑熵是否完全一致,例如映射笛卡尔积的点态原像熵的可加性等.本文将把环面自映射笛卡尔积的点态原像熵的可加性,推广到紧幂零流形自映射的情形. 相似文献
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动力系统中拓扑熵的研究 总被引:1,自引:0,他引:1
自从1965年R.L.Adler,A.G.Konheim和M.H.McAndrew对于紧致拓扑空间上的连续自映射,引进拓扑熵,1971年R.Bowen对度量空间上的一致连续自映射又重新定义以来,拓扑熵的概念逐渐深入到动力系统的研究工作中,并带来了值得注意的影响.有关的文章,据笔者所知已达数十篇之多.它之所以受到不少研究者的重视,正如廖山涛教授指出的,“在微分动力体系的探讨中一个有意义的一般问题是,寻找M~n上的结构稳定系统在拓扑等价意义下的数值不变量”.目下知道的数值不变量甚少,而拓扑熵就是这样一个数值不变量(见§2定理1). 相似文献
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We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow. 相似文献
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In this paper, we introduce the topological entropy of a free semigroup action generated by proper maps, which extends the notions of the topological entropy of the free semigroup actions defined by Bufetov in 1999 and topological entropy of the proper maps defined by Patrão in 2010. We then give some properties of these notions and discuss the relations between them. We also give a partial variational principle for locally compact separable metric spaces. Moreover, the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given. These results extend the results obtained by Patrão, Bufetov and Lin, Ma and Wang in 2018. 相似文献
13.
Definition of Measure-theoretic Pressure Using Spanning Sets 总被引:3,自引:0,他引:3
We introduce a new definition of measure–theoretic pressure for ergodic measures of continuous maps on a compact metric space.
This definition is similar to those of topological pressure involving spanning sets. As an application, for C
1+
α
(α > 0) diffeomorphisms of a compact manifold, we study the relationship between the measure–theoretic pressure and the periodic
points.
Project Supported by National Natural Science Foundation of China 相似文献
14.
《复变函数与椭圆型方程》2012,57(5):319-321
Let X be a topological space whose topology may be defined by a complete metric d. Taking all such metrics d we define a universal complex structure on X. For this complex structure the sheaf of germs of holomorphic functions on X coincides with the sheaf of germs of continuous functions on X, and hence the theories of topological and holomorphic vector bundles on X are the same. 相似文献
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D. Dryanov 《Constructive Approximation》2009,30(1):137-153
Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric
spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with
accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L
p
-metric proposed by A.I. Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L
p
-metric is ε
−1/2, ε→0+.
相似文献
16.
In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence
where Ad = (I + A) (I − A)−1.
We show that if A and A−1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then Ad is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of Ad is equivalent to the uniform boundedness of the semigroup generated by A. 相似文献
17.
Eli Glasner Michael Megrelishvili Vladimir V. Uspenskij 《Israel Journal of Mathematics》2008,164(1):317-332
When a topological group G acts on a compact space X, its enveloping semigroup
E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X
X
. Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d
G
(x, y) ≔ sup{d(gx, gy): g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following:
the enveloping semigroup E(X) is metrizable. 相似文献
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In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T
2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is
the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase
space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties
similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the
direction of geodesics lifted onto the universal covering
\mathbbR2{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient
conditions for vanishing topological entropy involving intersection properties of single geodesics on T
2. 相似文献
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We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g
-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank
h
X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank
h
conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H
n
with .
Submitted: February 2001, Revised: October 2001. 相似文献
20.
Wang Yangeng 《数学学报(英文版)》1997,13(3):333-336
The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel
2 ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R
n,Rm) is homomorphic to Hilbert spacel
2.
This research is supported by the Science Foundation of Shanxi Province's Scientific Committee 相似文献