Technical Developments of DA on $\mathbb{T}^3$

We constructed a DA on $\mathbb{T}^3$, which complements the work of Gan, Li, Viana, and Yang (\cite{GanLiVianaYang2021}) by providing an example of a $C^\infty$-diffeomorphism with partial volume expansion, where $\dim(E^{cs}) = 2$. In contrast to their work, in \cite{GanLiVianaYang2021}, they provided an example of a non-invertible embedding in the case $\dim(E^{cs}) = 2$. The inverse map of the DA we constructed has a mostly expanding center (\cite{AnderssonVasquez2018}). Using a similar approach, we can also construct a (nontrivial) mixed center (\cite{MiCaoYang2017, MiCao2021}).