Fix a nontrivial interval X ⊂ R X \subset {\mathbf {R}} and let f ∈ C 1 ( X , X ) f \in {C^1}(X,\,X) be a chaotic mapping. We denote by A ∞ ( f ) {A_\infty }(f) the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of f f or to a subset of the absorbing boundary of X X for f f . A. We assume that f f satisfies the following conditions: (1) the set of asymptotically stable periodic points for f f is compact (an empty set is allowed), and (2) A ∞ ( f ) A{\,_\infty }(f)\, is compact, f f is expanding on A ∞ ( f ) {A_\infty }(f) . Then we can associate a matrix A f {A_f} with entries either zero or one to the mapping f f such that the number of periodic points for f f with period n n is equal to the trace of the matrix [ A f ] n {\left [ {{A_f}} \right ]^n} ; furthermore the zeta function of f f is rational having the eigenvalues of A f {A_f} as poles. B. We assume that f ∈ C 3 ( X , X ) f \in {C^3}(X,\,X) such that: (1) the Schwarzian derivative of f f is negative, and (2) the closure of A ∞ ( f ) {A_\infty }(f) is compact and f ′ ( x ) ≠ 0 f’ (x) \ne 0 for all x x in the closure of A ∞ ( f ) {A_\infty }(f) . Then we obtain the same result as in A.