On the plaque topological stability of partially hyperbolic diffeomorphisms
L. Li, C. A. Morales, B. Shin
TL;DR
This work extends stability concepts from hyperbolic dynamics to partially hyperbolic systems by introducing plaque topological stability relative to the center foliation. It proves that dynamically coherent plaque expansive partially hyperbolic diffeomorphisms are plaque topologically stable with respect to their center foliation, and shows that, for expansive maps topologically stable w.r.t. a center foliation, the center chain recurrent set $CR^c(f)$ is contained in the closure of center periodic points $\overline{Per^c(f)}$, with $Per^c(f)\subset\Omega^c(f)$. The approach develops a framework of $\mathfrak{F}$-valued set-valued maps and a foliation-aware notion of continuity and shadowing to bridge perturbations with center-leaf behavior. These results generalize classical stability phenomena beyond uniform hyperbolicity and illuminate how center dynamics governs periodic and recurrent structures.
Abstract
We prove that every dynamically coherent plaque expansive partially hyperbolic diffeomorphism is topologically stable with respect to the central foliation (in short, {\em plaque topologically stable}). Next, we study partially hyperbolic diffeomorphisms that are both expansive and topologically stable with respect to a central foliation. We show that the center chain recurrent set for such diffeomorphisms belongs to the closure of the center periodic points.