Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics (Q2813941)

Let \(f:M\longrightarrow M\) be a partially hyperbolic \(C^1\) diffeomorphism with a uniformly compact \(f\)-invariant center foliation \(\mathcal{F}^c\). The authors investigate the topological properties of \(f\) and of its quotient map \(F: M/\mathcal{F}^c \longrightarrow M/\mathcal{F}^c\), and give the following results:NEWLINENEWLINE(1) \(f\) is dynamically coherent, i.e., \(E^{cu}:=E^c\oplus E^u\), \(E^{c}\), and \(E^{cs}:=E^c\oplus E^s\) are integrable, and everywhere tangent to \(\mathcal{W}^{cu}\) (which is called the \textit{center-unstable foliation}), \(\mathcal{W}^{c}\) and \(\mathcal{W}^{cs}\) (which is called the \textit{center-stable foliation}), respectively; and \(\mathcal{W}^c\) and \(\mathcal{W}^u\) are subfoliations of \(\mathcal{W}^{cu}\), while \(\mathcal{W}^c\) and \(\mathcal{W}^s\) are subfoliations of \(\mathcal{W}^{cs}\).NEWLINENEWLINE(2) Further, the induced homeomorphism \(F: M/\mathcal{F}^c \longrightarrow M/\mathcal{F}^c\) on the quotient space of the center foliation has the shadowing property, i.e., for every \(\varepsilon>0\) there exists \(\delta>0\) such that every \(\delta\)-pseudo orbit of center leaves is \(\varepsilon\)-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, they prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics.NEWLINENEWLINE(3) Some other interesting properties of the quotient dynamics, such as plaque expansivity and non-compactness of center-stable and center-unstable leaves for \(f\), are discussed.