Searcharxiv⌕ Search

arXiv subjects

Arezoo Hosseini

Publications and source records attributed to Arezoo Hosseini.

8 recordsLinked to original sources
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

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↗
arXiv2010

Algebraic entropy of shift endomorphisms on abelian groups

For every finite-to-one map $λ:Γ\toΓ$ and for every abelian group $K$, the generalized shift $σ_λ$ of the direct sum $\bigoplus_ΓK$ is the endomorphism defined by $(x_i)_{i\inΓ}\mapsto(x_{λ(i)})_{i\inΓ}$. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of $K$, but mainly on the function $λ$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.

math.GR↗