  Simultaneous tiling for several different translational sets has been studied
rather extensively, particularly in connection with the Steinhaus problem. The
study of orthonormal wavelets in recent years, particularly for arbitrary
dilation matrices, has led to the study of multiplicative tilings by the powers
of a matrix. In this paper we consider the following simultaneous tiling
problem: Given a lattice in $\L\in \R^d$ and a matrix $A\in\GLd$, does there
exist a measurable set $T$ such that both $\{T+\alpha: \alpha\in\L\}$ and
$\{A^nT: n\in\Z\}$ are tilings of $\R^d$? This problem comes directly from the
study of wavelets and wavelet sets. Such a $T$ is known to exist if $A$ is
expanding. When $A$ is not expanding the problem becomes much more subtle.
Speegle \cite{Spe03} exhibited examples in which such a $T$ exists for some
$\L$ and nonexpanding $A$ in $\R^2$. In this paper we give a complete solution
to this problem in $\R^2$.
