# American Institute of Mathematical Sciences

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
 [1] James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209 [2] Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363 [3] Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283 [4] Helge Krüger. Asymptotic of gaps at small coupling and applications of the skew-shift Schrödinger operator. Conference Publications, 2011, 2011 (Special) : 874-880. doi: 10.3934/proc.2011.2011.874 [5] J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397 [6] Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362 [7] Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305 [8] Emile Franc Doungmo Goufo, Melusi Khumalo, Patrick M. Tchepmo Djomegni. Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 663-682. doi: 10.3934/dcdss.2020036 [9] Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015 [10] Daniel Gonçalves, Marcelo Sobottka. Continuous shift commuting maps between ultragraph shift spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1033-1048. doi: 10.3934/dcds.2019043 [11] Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333 [12] Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036 [13] Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012 [14] Yaling Cui, Srdjan D. Stojanovic. Equity valuation under stock dilution and buy-back. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1809-1829. doi: 10.3934/dcdsb.2012.17.1809 [15] Leonid A. Bunimovich. Chaotic and nonchaotic mushrooms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 63-74. doi: 10.3934/dcds.2008.22.63 [16] Piotr Oprocha. Coherent lists and chaotic sets. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 797-825. doi: 10.3934/dcds.2011.31.797 [17] Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014 [18] José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269 [19] Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35 [20] Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483

2018 Impact Factor: 1.143