Mathematics and Statistics Vol. 8(5), pp. 535 - 541
DOI: 10.13189/ms.2020.080506
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-action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive

Nor Syahmina Kamarudin , Syahida Che Dzul-Kifli ^*
Faculty of Science and Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Malaysia

ABSTRACT

The dynamics of a multidimensional dynamical system may sometimes be inherited from the dynamics of its classical dynamical system. In a multidimensional case, we introduce a new map called a -action on space X induced by a continuous map as such that, where, and is a map of the form . We then look at how topological transitivity of f effects the behaviour of k-type transitivity of the -action, . To verify this, we look specifically at spaces called 1-step shifts of finite type over two symbols which are equipped with a map called the shift map, . We apply some topological theories to prove the -action on 1-step shifts of finite type over two symbols induced by the shift map, is k-type transitive for all whenever is topologically transitive. We found a counterexample which shows that not all maps are k-type transitive for all . However, we have also found some sufficient conditions for k-type transitivity for all. In conclusions, the map on 1-step shifts of finite type over two symbols induced by the shift map is k-type transitive for all whenever either the shift map is topologically transitive or satisfies the sufficient conditions. This study helps to develop the study of k-chaotic behaviours of -action on the multidimensional dynamical system, contributions, and its application towards symbolic dynamics.

KEYWORDS
-action, Shift of Finite Type, Topologically Transitive, k-type Transitive

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Nor Syahmina Kamarudin , Syahida Che Dzul-Kifli , "-action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive," Mathematics and Statistics, Vol. 8, No. 5, pp. 535 - 541, 2020. DOI: 10.13189/ms.2020.080506.

(b). APA Format:
Nor Syahmina Kamarudin , Syahida Che Dzul-Kifli (2020). -action Induced by Shift Map on 1-Step Shift of Finite Type over Two Symbols and k-type Transitive. Mathematics and Statistics, 8(5), 535 - 541. DOI: 10.13189/ms.2020.080506.