We show that the smooth \(4\)-manifold \(M\) obtained by attaching a \(2\)-handle to \(B^4\) along a certain knot \(K\subset \partial B^4\) admits infinitely many absolutely exotic copies \(M_n\), \(n=0,1,2..\), such that each copy \(M_n\) is obtained by attaching \(2\)-handle to a fixed compact smooth contractible manifold \(W\) along the iterates \(f^{n}(c)\) of a knot \(c\subset \partial W\) by a diffeomorphism \(f:\partial W \to \partial W\). This generalizes the example in author's 1991 paper, which corresponds to \(n=1\) case in [A1].