arXiv subjects

Fatemeh Ebrahimifar

Publications and source records attributed to Fatemeh Ebrahimifar.

6 recordsLinked to original sources

arXiv2021

On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems

In the following text, for finite discrete $X$ with at least two elements, nonempty countable $Γ$, and $φ:Γ\toΓ$ we prove the generalized shift dynamical system $(X^Γ,σ_φ)$ is densely chaotic if and only if $φ:Γ\toΓ$ does not have any (quasi-)periodic point. Hence the class of all densely chaotic generalized shifts on $X^Γ$ is intermediate between the class of all Devaney chaotic generalized shifts on $X^Γ$ and the class of all Li-Yorke chaotic generalized shifts on $X^Γ$. In addition, these inclusions are proper for infinite countable $Γ$. Moreover we prove $(X^Γ,σ_φ)$ 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 $φ:Γ\toΓ$ has at least one non-quasi-periodic point.

math.DS↗

arXiv2020

On special subgroups of fundamental group

Suppose $α$ is a nonzero cardinal number, $\mathcal I$ is an ideal on arc connected topological space $X$, and ${\mathfrak P}_{\mathcal I}^α(X)$ is the subgroup of $π_1(X)$ (the first fundamental group of $X$) generated by homotopy classes of $α\frac{\mathcal I}{}$loops. The main aim of this text is to study ${\mathfrak P}_{\mathcal I}^α(X)$s and compare them. Most interest is in $α\in\{ω,c\}$ and $\mathcal I\in\{\mathcal P_{fin}(X),\{\varnothing\}\}$, where $\mathcal P_{fin}(X)$ denotes the collection of all finite subsets of $X$. We denote ${\mathfrak P}_{\{\varnothing\}}^α(X)$ with ${\mathfrak P}^α(X)$. We prove the following statements: $\bullet$ for arc connected topological spaces $X$ and $Y$ if ${\mathfrak P}^α(X)$ is isomorphic to ${\mathfrak P}^α(Y)$ for all infinite cardinal number $α$, then $π_1(X)$ is isomorphic to $π_1(Y)$; $\bullet$ there are arc connected topological spaces $X$ and $Y$ such that $π_1(X)$ is isomorphic to $π_1(Y)$ but ${\mathfrak P}^ω(X)$ is not isomorphic to ${\mathfrak P}^ω(Y)$; $\bullet$ for arc connected topological space $X$ we have ${\mathfrak P}^ω(X)\subseteq{\mathfrak P}^c(X) \subseteqπ_1(X)$; $\bullet$ for Hawaiian earring $\mathcal X$, the sets ${\mathfrak P}^ω({\mathcal X})$, ${\mathfrak P}^c({\mathcal X})$, and $π_1({\mathcal X})$ are pairwise distinct. So ${\mathfrak P}^α(X)$s and ${\mathfrak P}_{\mathcal I}^α(X)$s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

math.AT↗

arXiv2018

Possible heights of Alexandroff square transformation groups

In the following text we compute possible heights of $\mathbb A$ (Alexandroff square), $\mathbb O$ (unit square $[0,1]\times[0,1]$ with lexicographic order topology) and $\mathbb U$ (unit square $[0,1]\times[0,1]$ with induced topology of Euclidean plane). We prove $P_h(\mathbb{A})=\{n:n\geq5\}\cup\{+\infty\}$, $P_h(\mathbb{O})=\{n:n\geq4\}\cup\{+\infty\}$, $P_h(\mathbb{U})=\{n:n\geq1\}\cup\{+\infty\}$ (where for topological space $X$, by $P_h(X)$ we mean the collection of heights of transformation groups with phase space $X$. In this way we also prove that there is not any topological transitive (resp. Devaney chaotic) Alexandroff square transformation group.

math.GN↗

arXiv2018

Is there any nontrivial compact generalized shift operator on Hilbert spaces?

In the following text for cardinal number $τ>0$, and self--map $φ:τ\toτ$ we show the generalized shift operator $σ_φ(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_φ((x_α)_{α<τ})=(x_{φ(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $φ:τ\toτ$ is bounded and in this case $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite.

math.FA↗

arXiv2017

On generalized shift transformation semigroups

In the following text we prove that for finite discrete $X$ with at least two elements and infinite $Γ$, the generalized shift transformation semigroup $({\mathcal S},X^Γ)$ is equicontinuous (resp. has at least an equicontinuous point, is not sensitive) if and only if for all $w\inΓ$, $\{φ(w):σ_φ\in{\mathcal S}\}$ is finite. We continue our study regarding distality and expansivity of $({\mathcal S},X^Γ)$.

math.DS↗

arXiv2017

Indicator sequences and indicator topologies of Fort transformation groups

In the following text we prove that there exists a Fort transformation group with indicator sequence $(p_0,\ldots,p_n)$ if and only if $0=p_0\leq p_1\leq\cdots\leq p_n=1$, moreover we characterize all possible indicator topological spaces of Fort transformation groups.The text will study indicator sequences and indicator topologies of Fort transformation semigroups too.

math.GN↗