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SOME NONROBUST BERNOULLI-SHIFT RULES

LIN CHEN

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

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FANGYUE CHEN

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, P. R. China

Author for correspondence.

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WEIFENG JIN

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

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FANGFANG CHEN

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

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, and

GUANRONG CHEN

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

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https://doi.org/10.1142/S0218127409024840Cited by:4 (Source: Crossref)

Abstract

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

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