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Fatemah Ayatollah Zadeh Shirazi

Publications and source records attributed to Fatemah Ayatollah Zadeh Shirazi.

At least 19 recordsLinked to original sources
arXiv2025

Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example

The main aim of this paper is extending the concept of scambled pair and Li--Yorke chaos to non--uniform compact dynamical systems. We show for finite (compact Alexandroff) topological space $X$ with at least two elements the following statements are equivalent: $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ is Li--Yorke chaotic, $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one scrambled pair, $\bullet$ one--sided shift $σ:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one non--asymptotic pair, $\bullet$ there exists $a,b\in X$ such that $\overline{\{a\}}\cap\overline{\{b\}}=\varnothing$, $\bullet$ $\bigcap\{\overline{\{a\}}:a\in X\}=\varnothing$.

math.DS↗
arXiv2025

Closed graph property in Alexandroff spaces

In the following text we show if $X$ is an Alexandroff space, then $f:X\to Y$ has closed graph if and only if it has constant closed value on each connected component of $X$. Moreover, if $X$ an Alexandroff space and $f:X\to Y$ has closed graph, then $f:X\to Y$ is continuous. As a matter of fact, the number of maps which have closed graph from Alexandroff space $X$ to a topological space $Y$ depends just on the the number of connected components of $X$ and the number of closed points of $Y$.

math.GN↗
arXiv2025

Closed graph property and Khalimsky spaces

In the following text for Khalimsky $n-$dimensional space $\mathcal{K}^n$ we show self--map $f:\mathcal{K}^n\to\mathcal{K}^n$ has closed graph if and only if there exist integers $λ_1,\ldots,λ_n$ such that $f$ is a constant map with value $(2λ_1,\cdots,2λ_n)$. We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.

math.GN↗
arXiv2024

Set-theoretical entropies of weighted generalized shifts

In this paper for a finite field $F$, a nonempty set $Γ$, a self--map $φ:Γ\toΓ$ and a weight vector $\mathfrak{w}\in F^Γ$, we show that the set--theoretical entropy of the weighted generalized shift $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is either zero or $+\infty$, moreover it is equal to zero if and only if $σ_{φ,\mathfrak{w}}$ is quasi--periodic. On the other hand after characterizing all conditions under which $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ is of finite fibre, we show that the contravariant set--theoretical entropy of a finite fibre $σ_{φ,\mathfrak{w}}:F^Γ\to F^Γ$ depends only on $φ$ and $\rm{supp}(\mathfrak{w})$. In final sections we study the restriction of $σ_{φ,\mathfrak{w}}$ to the direct sum $\mathop{\bigoplus}\limits_ΓF$.

math.GM↗
arXiv2023

Uniformizable functional Alexandroff spaces

In the following text we show that the Alexandroff space $X$ is uniformizable if and only if the collection of all smallest neighbourhoods is a partition of $X$. Moreover the Alexandroff space $X$ is uniformizable and functional Alexandroff ($k-$primal) if and only if the collection of all smallest neighbourhoods is a partition of $X$ into its finite subsets.

math.GN↗
arXiv2022

On pseudo-codecomposition of a transformation group

In the following text we introduce the concept of pseudo-codecomposition of a transformation group, also we show the collection of all transformation groups pseudo-codecomposable to distal ones is a proper intermediate class of the class of all transformation groups and the class of all transformation groups codecomposable to distal ones.

math.GT↗
arXiv2022

Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts

In this paper, for finite discrete field $F$, nonempty set $Γ$, weight vector $\mathfrak{w}=({\mathfrak w}_α)_{α\inΓ}\in F^Γ$ and weighted generalized shift $σ_{φ,{\mathfrak w}}:F^Γ\to F^Γ$, we find necessary and sufficient conditions for uniform dynamical system $(F^Γ,σ_{φ,{\mathfrak w}})$ to be Li--Yorke chaotic. Next we find necessary and sufficient conditions for $(F^Γ,σ_{φ,{\mathfrak w}})$ to be Devaney chaotic.

math.DS↗
arXiv2021

Possible heights of graph transformation groups

In the following text we prove that for all finite $p\geq0$ there exists a topological graph $X$ such that $\{p,p+1,p+2,\ldots\}\cup\{+\infty\}$ is the collection of all possible heights for transformation groups with phase space $X$. Moreover for all topological graph $X$ with $p$ as height of transformation group $(Homeo(X),X)$, $\{p,p+1,p+2,\ldots\}\cup\{+\infty\}$ again is the collection of all possible heights for transformation groups with phase space $X$.

math.GT↗
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↗
arXiv2021

Generalized shifts through derivations' concept in $\ell^p(τ)$ spaces

In the following text for $p\in[1,\infty]$, nonzero cardinal number $τ$, self--map $φ:τ\toτ$ if there exists $N\in\mathbb{N}$ such that $φ^{-1}(α)$ has at most $N$ elements for each $α<τ$, and operators $ψ,λ:\ell^pτ)\to\ell^p(τ)$ we prove the generalized shift $\mathop{σ_φ\restriction_{\ell^p(τ)}:\ell^p(τ)\to\ell^p(τ)\:\:\:\:\:\:\:\:\:}\limits_{\:\:\:\:\:\:\:\:\: (x_α)_{α<τ}\mapsto (x_{φ(α)})_{α<τ}}$: $\bullet$ is a $(ψ,λ)-$derivation if and only if there exists $\mathsf{r}\in{\mathbb C}^τ$ with $ψ={\mathsf r}σ_φ\restriction_{\ell^p(τ)}$ and $λ=((1)_{α<τ}-{\mathsf r})σ_φ\restriction_{\ell^p(τ)}$, $\bullet$ is a $ψ-$derivation if and only if $ψ=\frac12σ_φ\restriction_{\ell^p(τ)}$, $\bullet$ is not a (Jordan, Jordan triple) derivation, $\bullet$ is a generalized (Jordan, Jordan triple) derivation if and only if $φ=id_τ$.

math.FA↗
arXiv2021

On Li--Yorke chaotic transformation groups modulo an ideal

In the following text we introduce the notion of chaoticity modulo an ideal in the sense of Li-Yorke in topological transformation semigroups and bring some of its elementary properties. We continue our study by characterizing a class of abelian infinite Li-Yorke chaotic Fort transformation groups and show all of the elements of the above class is co-decomposable to non-Li-Yorke chaotic transformation groups.

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↗
arXiv2020

Distributional chaotic generalized shifts

Suppose $X$ is a finite discrete space with at least two elements, $Γ$ is a nonempty countable set, and consider self--map $φ:Γ\toΓ$. We prove that the generalized shift $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})=(x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is: $\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $φ:Γ\toΓ$ has at least a non-quasi-periodic point, $\bullet$ dense distributional chaotic if and only if $φ:Γ\toΓ$ does not have any periodic point, $\bullet$ transitive distributional chaotic if and only if $φ:Γ\toΓ$ is one--to--one without any periodic point. We complete the text by counterexamples.

math.DS↗
arXiv2019

On proximal relations in transformation semigroups arising from generalized shifts

For a finite discrete topological space $X$ with at least two elements, a nonempty set $Γ$, and a map $φ:Γ\toΓ$, $σ_φ:X^Γ\to X^Γ$ with $σ_φ((x_α)_{α\inΓ})= (x_{φ(α)})_{α\inΓ}$ (for $(x_α)_{α\inΓ}\in X^Γ$) is a generalized shift. In this text for $\mathcal{S}=\{σ_ψ:ψ\inΓ^Γ\}$ and $\mathcal{H}=\{σ_ψ: Γ\mathop{\rightarrow}\limits^ψΓ$ is bijective$\}$ we study proximal relations of transformation semigroups $(\mathcal{S},X^Γ)$ and $(\mathcal{H},X^Γ)$. Regarding proximal relation we prove: \[P({\mathcal S},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \existsβ\inΓ\:(x_β=y_β)\}\] and $P({\mathcal H},X^Γ)\subseteq \{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{β\inΓ:x_β=y_β\}$ is infinite~$\}\cup\{ (x,x):x\in \mathcal{X}\}$. \\ Moreover, for infinite $Γ$, both transformation semigroups $({\mathcal S},X^Γ)$ and $({\mathcal H},X^Γ)$ are regionally proximal, i.e., $Q({\mathcal S},X^Γ)=Q({\mathcal H},X^Γ)=X^Γ\times X^Γ$, also for sydetically proximal relation we have $L({\mathcal H},X^Γ)=\{((x_α)_{α\inΓ},(y_α)_{α\inΓ}) \in X^Γ\times X^Γ: \{γ\inΓ:x_γ\neq y_γ\}$ is finite$\}$.

math.DS↗
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

Specification properties on uniform spaces

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space $X$ with at least two elements, a nonempty set $Γ$ and a self--map $φ:Γ\toΓ$ the generalized shift dynamical system $(X^Γ,σ_φ)$: \begin{itemize} \item has (almost) weak specification property if and only if $φ:Γ\toΓ$ does not have any periodic point, \item has (uniform) stroboscopical property if and only if $φ:Γ\toΓ$ is one-to-one. \end{itemize}

math.DS↗
arXiv2018

Set-theoretical entropies of Euler's totient function and other number theoretical special functions

In the following text we show set--theoretical entropy of Euler's totient function and contravariant set--theoretical entropy of Dedekind psi function are zero. Also contravariant set--theoretical entropy of Euler's totient function and set--theoretical entropy of Dedekind psi function are $+\infty$. We pay attention to some of the other number theoretical special functions too. We continue our studies on Alexandroff topologies induced by Euler's totient function and Dedekind psi function.

math.NT↗