Partially Hyperbolic Dynamics on ${\mathbb{T}}^4$: Existence of Compact Center-Unstable Leaves

We show that for $n\ge 2$, if a partially hyperbolic diffeomorphism $f:{\mathbb{T}}^{n+1}\to {\mathbb{T}}^{n+1}$ with $\dim E^s=\dim E^c=1$ has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a transverse closed curve in the universal cover. Also, if $f$ is leaf conjugate to its linear part, it has no compact incompressible center-unstable submanifold. In particular, by the incompressibility result we obtained on Anosov tori, the incompressibility assumptions can be removed when $f$ is defined on ${\mathbb{T}}^4$.