An answer to Hammerlindl's question on strong unstable foliations

Abstract

Let $f:{\mathbb{T}}^3\to {\mathbb{T}}^3$ be a partially hyperbolic diffeomorphism on the 3-torus ${\mathbb{T}}^3$. In his thesis, Hammerlindl proved that for lifted center foliation ${\mathcal{F}}_f^c$, there exists $R>0$, such that for any $x\in {\mathbb{R}}^3$, ${\mathcal{F}}_f^c(x)\subset B_R(x+E^c)$, where ${\mathbb{R}}^3=E^s\oplus E^c\oplus E^u$ is the partially hyperbolic splitting of the linear model of $f$. The same is true for the lifted center-stable and center-unstable foliations. Then he asked if the this property is true for strong stable and strong unstable foliations. In this note, we give a negative answer to Hammerlindl's question.