-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 · Mathematics and Statistics · 2020
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.