  A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to
be an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus
  \[
  \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n
\right\} \subseteq \mathbb{C}^n \qquad \quad (0<r<1)
  \]
  is a spectral set for $(T_1, \dotsc, T_n)$. We find characterizations for the
$\mathbb A_r^n$-unitaries and $\mathbb A_r^n$-isometries and decipher their
structures. We find Wold type decompositions for any number of commuting and
doubly commuting $\mathbb A_r$-isometries. Then we generalize these results to
any family of commuting and doubly commuting $\mathbb A_r$-contractions.
