Various notions of topological transitivity in non-autonomous and generic dynamical systems
Chiranjeevi Perikala, Rameshwari Gupta
TL;DR
The paper examines multiple notions of topological transitivity in non-autonomous discrete dynamical systems (NDDS) and generic dynamical systems (GDS) on compact metric spaces. It introduces extended transitivity via the extended orbit $J(y)$, establishes a network of equivalent conditions for TT, ST, VST, extended variants, exact variants, and mixing, and analyzes how these properties are preserved under (strong) semiconjugacy and finite rearrangements. A key contribution is linking NDDS to an associated GDS by selecting suitable generating families $\mathcal{F}$ and showing how transitivity notions correspond across frameworks. The results yield a coherent hierarchy of transitivity notions (e.g., ${\rm LEO} \Rightarrow {\rm VST} \Rightarrow {\rm ST} \Rightarrow {\rm TT}$) and provide practical criteria for verifying transitivity properties in both NDDS and GDS, with implications for their structural robustness under conjugacy and rearrangements.
Abstract
We consider two types of dynamical systems namely non-autonomous discrete dynamical systems(NDDS) and generic dynamical systems(GDS). In both of them, we study various notions of transitivity. We give many equivalent conditions for each of these notions and present the implications among these in NDDS and GDS. For a given NDDS, we associate a GDS and discuss whether if the given NDDS has a particular variation of transitivity then the associated GDS also has such a variation and vice versa.