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2000•International Journal of Bifurcation and ChaosOpen access

CHAOTIC PROPERTIES OF SUBSHIFTS GENERATED BY A NONPERIODIC RECURRENT ORBIT

Xinchu Fu, Yibin Fu, Jinqiao Duan, Robert S. MacKay

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Abstract

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

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The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

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Available abstract

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain nonperiodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a nonperiodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

Key concepts: Topological entropy, Subshift of finite type, Chaotic, Mathematics, Symbolic dynamics, Orbit (dynamics), Periodic point, Periodic orbits

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