Malaysian Journal of Mathematical Sciences, September 2022, Vol. 16, No. 3
On A Class Between Devaney Chaotic and Li-Yorke Chaotic Generalized Shift Dynamical Systems
Ayatollah Zadeh Shirazi, F., Ebrahimifar, F., Hagh Jooyan, M., and Hosseini, A.
Corresponding Email: a.hosseini@cfu.ac.ir
Received date: 12 September 2020
Accepted date: 20 June 2022
Abstract:
In the following text,
for finite discrete $X$ with at least two elements, nonempty countable $\Gamma$, and $\varphi:\Gamma\to\Gamma$
we prove the generalized shift dynamical system $(X^\Gamma,\sigma_\varphi)$ is densely chaotic if and only if
$\varphi:\Gamma\to\Gamma$ does not have any (quasi--)periodic point.
Hence the class of all densely chaotic generalized shifts on $X^\Gamma$ is intermediate between
the class of all Devaney chaotic generalized shifts on $X^\Gamma$ and
the class of all Li--Yorke chaotic generalized shifts on $X^\Gamma$.
In addition, these inclusions are proper for infinite countable $\Gamma$.
Moreover we prove $(X^\Gamma,\sigma_\varphi)$ is
Li--Yorke sensitive (resp. sensitive, strongly sensitive, asymptotic sensitive,
syndetically sensitive, cofinitely sensitive, multi--sensitive, ergodically sensitive, spatiotemporally chaotic,
Li--Yorke chaotic) if and only if $\varphi:\Gamma\to\Gamma$ has at least one non--quasi--periodic point.
Keywords: asymptotic sensitive; densely chaotic; Li-Yorke sensitive; spatiotemporally chaotic; strongly sensitive; generalized shift
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