A method for determination and two methods for approximation of the domain of attraction D a ( 0 ) of the asymptotically stable zero steady state of an autonomous, ℝ -analytical, discrete dynamical system are 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 converge to the Lyapunov function V on D a ( 0 ) . Each V p defines an estimate N p of D a ( 0 ) . For any x ∈ D a ( 0 ) , there exists an estimate N p x which contains x . The second method of approximation uses a ball B ( R ) ⊂ D a ( 0 ) which generates the sequence of estimates M p = f − p ( B ( R ) ) . For any x ∈ D a ( 0 ) , there exists an estimate M p x which contains x . The cases ‖ ∂ 0 f ‖ < 1 and ρ ( ∂ 0 f ) < 1 ≤ ‖ ∂ 0 f ‖ are treated separately because significant differences occur.