  Motivated by recent progress in operator representation of frames, we
investigate the frames of the form $ \{T^n \varphi\}_{n\in I}$ for $
I=\mathbb{N}, \mathbb{Z} $, and answer questions about representations,
perturbations and frames induced by the action of powers of bounded linear
operators. As a particular case, we discuss problems concerning representation
of frames in terms of iterations of the mixed frame operators. As our another
contribution, we consider frames of the form $ \{a_n T^n
\varphi\}_{n=0}^{\infty} $ for some non-zero scalars $ \{a_n\}_{n=0}^{\infty}
$, and we obtain some new results in dynamical sampling. Finally, we will
present some auxiliary results related to the perturbation of sequences of the
form $ \{T^n \varphi\}_{n=0}^{\infty}$.
