Chaotic maps with rational zeta function

Abstract:

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