Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations
Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations
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DOI：
10.3934/dcds.2007.19.103
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发表时间：
2007-06
影响因子：
1.1
通讯作者：
Wei Lin;Jianhong Wu;Guanrong Chen
中科院分区：
文献类型：
--
作者：
Wei Lin;Jianhong Wu;Guanrong Chen
在本文中，我们试图澄清一个公开的问题，有关的一个概括的反弹排斥。构造一个半共轭从有限积的变换$f：\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$上的不变集$\Lambda$上的有限类型的子移位的$w$-符号空间，我们表明，相应的变换与广义snap-back排斥$\mathbb{R}^{n}$上表现出混沌动力学的意义下，有一个积极的拓扑熵。导致这一结论的论证还表明，某种退化变换，允许一个点的不稳定流形的排斥映射回到排斥，有积极的拓扑熵的轨道上的不变集。此外，我们提出了两个可行的充分条件，获得一个不稳定的流形。最后，我们提供了两个说明性的例子来说明混沌退化变换是无所不在的。
In this paper, we attempt to clarify an open problem related to a generalization of the snap-back repeller. Constructing a semi-conjugacy from the finite product of a transformation $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ on an invariant set $\Lambda$ to a sub-shift of the finite type on a $w$-symbolic space, we show that the corresponding transformation associated with the generalized snap-back repeller on $\mathbb{R}^{n}$ exhibits chaotic dynamics in the sense of having a positive topological entropy. The argument leading to this conclusion also shows that a certain kind of degenerate transformations, admitting a point in the unstable manifold of a repeller mapping back to the repeller, have positive topological entropies on the orbits of their invariant sets. Furthermore, we present two feasible sufficient conditions for obtaining an unstable manifold. Finally, we provide two illustrative examples to show that chaotic degenerate transformations are omnipresent.