Chaotic maps with rational zeta function

Abstract

ABSTRACT. Fix a nontrivial interval X C R and let / e C1(X,X) be a chaotic mapping. We denote by Aoo (/) the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of / or to a subset of the absorbing boundary of X for /.A. We assume that / satisfies the following conditions: (1) the set of asymptotically stable periodic points for / is compact (an empty set is allowed), and (2) AoAf) is compact, / is expanding on Aoc(f).Then we can associate a matrix Aj with entries either zero or one to the mapping / such that the number of periodic points for / with period n is equal to the trace of the matrix [Ay]"; furthermore the zeta function of / is rational having the eigenvalues of Af as poles.B. We assume that / 6 C3(X, X) such that: (1) the Schwarzian derivative of / is negative, and (2) the closure of Aoo(/) is compact and f'(x) ^ 0 for all x in the closure of Acx)(/). Then we obtain the same result as in A.