March  2007, 19(1): 103-119. doi: 10.3934/dcds.2007.19.103

Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations

1. 

Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

2. 

Department of Mathematics and Statistics, York University, Toronto, Canada M3J 1P3

3. 

Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong S.A.R., China

Received  August 2005 Revised  May 2007 Published  June 2007

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.
Citation: Wei Lin, Jianhong Wu, Guanrong Chen. Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 103-119. doi: 10.3934/dcds.2007.19.103
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