arXiv subjects

Shital H. Joshi

Publications and source records attributed to Shital H. Joshi.

3 recordsLinked to original sources

arXiv2025

Topological $(\mathscr{F},\mathscr{G})-$shadowing property

We define the concept of $(\mathscr{F},\mathscr{G})-$shadowing property on uniform space and say it as a topological $(\mathscr{F},\mathscr{G})-$shadowing property. We show that topological shadowing, topological $(\mathscr{D},\mathscr{F}_t)-$shadowing, topological $(\mathscr{F}_t,\mathscr{F}_t)-$shadowing and topological $(\mathbb{N},\mathscr{F}_t)-$shadowing are equivalent in compact chain recurrent dynamical system. We also prove that if minimal points of $f$ are not dense in $X$ then the system does not have $(\mathscr{P}(\mathbb{N}),\mathscr{F}_{ps})-$shadowing.

math.DS↗

arXiv2025

On stronger forms of Devaney chaos

We define and study stronger forms of Devaney chaos and name it as $\mathscr{F}-$Devaney chaos, where $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of maps which is $\mathscr{F}_t-$Devaney chaotic but not $\mathscr{F}_{cf}-$Devaney chaotic, $\mathscr{F}_s-$Devaney chaotic but neither $\mathscr{F}_t-$Devaney chaotic nor $\mathscr{F}_{cf}-$Devaney chaotic are discussed. Further, we show that for the maps on infinite metric space without isolated points, $\mathscr{F}-$sensitivity is a redundant condition in the definition $\mathscr{F}-$Devaney chaos. Here $\mathscr{F}=\mathscr{F}_s, \: \mathscr{F}_t, \: \mathscr{F}_{ts}$ or $\mathscr{F}_{cf}$. We also obtain conditions under which Devaney chaos implies $\mathscr{F}_s-$Devaney chaos or $\mathscr{F}_t-$Devaney chaos. Next, we define the concept of $\left(\mathscr{F}, \mathscr{G}\right)-P-$chaos and obtain conditions under which $\left(\mathscr{F}_1, \mathscr{G}_1\right)-P-$chaos implies $\mathscr{F}-$Devaney chaos for different families $\mathscr{F}_1$ and $\mathscr{G}_1$.

math.DS↗

arXiv2024

On Stronger Forms of Expansivity

We define the concept of stronger forms of positively expansive map and name it as $p \:\mathscr{F}-$expansive maps. Here $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of positively thick expansive and positively syndetic expansive maps are constructed here. Also, we obtain conditions under which a positively expansive map is positively co--finite expansive and positively syndetic expansive maps. Further, we study several properties of $p \:\mathscr{F}-$expansive maps. A characterization of $p \:\mathscr{F}-$expansive maps in terms of $p \:\mathscr{F}^*-$generator is obtained. Here $p \:\mathscr{F}^*$ is dual of $\mathscr{F}$. Considering $(\mathbb{Z},+)$ as a semigroup, we study $\mathscr{F}-$expansive homeomorphism, where $\mathscr{F}$ is a family of subsets of $\mathbb{Z} \setminus \{0\}$. We show that there does not exists an expansive homeomorphism on a compact metric space which is $\mathscr{F}_s-$expansive. Also, we study relation between $\mathscr{F}-$expansivity of $f$ and the shift map $σ_f$ on the inverse limit space.

math.DS↗