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首页> 外文期刊>Discrete and continuous dynamical systems >A SURVEY OF SOME ASPECTS OF DYNAMICAL TOPOLOGY: DYNAMICAL COMPACTNESS AND SLOVAK SPACES
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A SURVEY OF SOME ASPECTS OF DYNAMICAL TOPOLOGY: DYNAMICAL COMPACTNESS AND SLOVAK SPACES

机译：动态拓扑若干方面的调查：动态致密度和斯洛伐克空间

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The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dyna- mics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called "Dynamical Topology". As an illustration, some topolog- ical properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". A Slovak space, as a dynamical ana- logue of a rigid space, is a nontrivial compact metric space whose homeomor- phism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the col- lection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega- limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.
机译：一种调查可以以拓扑术语描述的动态特性的动态系统的区域是“拓扑Dyna-Mics”。研究可以以动态术语描述的空间和地图的拓扑特性是一个感觉相反的想法。该地区最近被称为“动态拓扑”。作为图示，描述了所有传递间隔图的空间的一些拓扑结构。对于由紧凑型度量空间给出的（离散的）动态系统，以及连续（调查）自我映射，我们调查了一些新概念的结果：“斯洛伐克空间”和“动态紧凑型”。作为刚性空间的动态ANA的斯洛伐克空间是一个非活动的紧凑型公制空间，其家用莫博物群是循环的，由最小的同胚术产生。动态紧致性是一种新的混沌动力学概念。一点的欧米茄限制集是动态系统理论中的基本概念，并且意味着“吸引”这一点的状态，同时在进行期间的状态。当相位空间紧凑时，它总是不动作。通过改变我们介绍了关于Furstenberg家族的欧米茄限制集的概念。如果对于任何点，则称为动态系统称为动态紧凑（关于Furstenberg家族），这种omega限制集是非空的。动态紧凑性的良好性质是，如果这个家庭具有有限的交叉点，则所有动态系统都相对于Furstenberg家族动态紧凑。

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