  In this paper, we propose an approach for computing invariant sets of
discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher
dimensional linear model. In particular, we focus on the \emph{maximal
admissible invariant set} contained in some given constraint set. For special
types of nonlinear systems, which can be exactly immersed into higher
dimensional linear systems with state transformations, invariant sets of the
original nonlinear system can be characterized using the higher dimensional
linear representation. For general nonlinear systems without the immersibility
property, \emph{approximate immersions} are defined in a local region within
some tolerance and linear approximations are computed by leveraging the
fixed-point iteration technique for invariant sets. Given the bound on the
mismatch between the linear approximation and the original system, we provide
an invariant inner approximation of the \emph{maximal admissible invariant set}
by a tightening procedure.
