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 $\mathcal{F}^c_f$, there exists $R>0$, such that for any $x\in \mathbb{R}^3$, ${\cal F}^c_f(x)\subset B_R (x+E^c)$, 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.