On stronger forms of Devaney chaos
Shital H. Joshi, Ekta Shah
TL;DR
The paper develops a framework to strengthen Devaney chaos via $\mathscr{F}$-sensitivity and $\mathscr{F}$-transitivity, defining $\mathscr{F}$-Devaney chaos and exploring its relation to classical Devaney chaos, shadowing, and P-chaos. It introduces $(\mathscr{F},\mathscr{G})$-P-chaos as a shadowing-based generalization and derives conditions under which it yields $\mathscr{F}$-Devaney chaos across different families; it also analyzes shift spaces, Sturmian and spacing shifts, and interval maps to illustrate when $\mathscr{F}$-Devaney chaos holds or fails. The results unify and extend known chaoticity criteria, showing, for instance, that $\mathscr{F}$-Devaney chaos follows from density of periodic points together with $\mathscr{F}$-transitivity in infinite spaces, and clarifying the hierarchy among $\mathscr{F}_s$, $\mathscr{F}_t$, $\mathscr{F}_{ts}$, $\mathscr{F}_{cf}$. The work provides both concrete examples and general theorems linking shadowing, transitivity, and sensitivity in a familywise chaos framework, with implications for transfer of chaos properties under conjugacy and for interval dynamics.
Abstract
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 $\mathscr{G}_1$.