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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Chaotic maps with rational zeta function


Author: H. E. Nusse
Journal: Trans. Amer. Math. Soc. 304 (1987), 705-719
MSC: Primary 58F13; Secondary 58F14, 58F20
MathSciNet review: 911091
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0911091-1
PII: S 0002-9947(1987)0911091-1
Keywords: Iteration of mappings, periodic points, semigroup of chaotic mappings, zeta function
Article copyright: © Copyright 1987 American Mathematical Society