  A method for determination and two methods for approximation of the domain of
attraction $D_{a}(0)$ of an asymptotically stable steady state of an
autonomous, $\mathbb{R}$-analytical, discrete system is presented. The method
of determination is based on the construction of a Lyapunov function $V$, whose
domain of analyticity is $D_{a}(0)$. The first method of approximation uses a
sequence of Lyapunov functions $V_{p}$, which converges to the Lyapunov
function $V$ on $D_{a}(0)$. Each $V_{p}$ defines an estimate $N_{p}$ of
$D_{a}(0)$. For any $x\in D_{a}(0)$ there exists an estimate $N_{p^{x}}$ which
contains $x$. The second method of approximation uses a ball $B(R)\subset
D_{a}(0)$ which generates the sequence of estimates $M_{p}=f^{-p}(B(R))$. For
any $x\in D_{a}(0)$ there exists an estimate $M_{p^{x}}$ which contains $x$.
The cases $\|\partial_{0}f\|<1$ and $\rho(\partial_{0}f)<1$ are treated
separately (even though the second case includes the first one) because
significant differences occur.
