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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.

References [Enhancements On Off] (What's this?)

  • [1] D. J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34 (1978), 687-691. MR 0475982 (57:15563)
  • [2] M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82-99. MR 0176482 (31:754)
  • [3] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391-398. MR 539925 (80m:58031)
  • [4] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725-747. MR 0277003 (43:2740)
  • [5] P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser, 1980. MR 613981 (82j:58078)
  • [6] J. Guckenheimer, On the bifurcations of maps of the interval, Invent. Math. 39 (1977), 165-178. MR 0438399 (55:11312)
  • [7] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. MR 0385028 (52:5898)
  • [8] J. Milnor and W. Thurston, On iterated maps of the interval. I and II, Mimeographed, Princeton Univ., 1977.
  • [9] M. Misiurewicz, Structure of mappings of an interval with zero entropy, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 5-16. MR 623532 (83j:58071)
  • [10] -, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 17-51. MR 623533 (83j:58072)
  • [11] Z. Nitecki, Topological dynamics on the interval, Ergodic Theory and Dynamical Systems. II, Progress in Math., 21, Birkhäuser, 1982, pp. 1-73. MR 670074 (84g:54051)
  • [12] H. E. Nusse, Chaos, yet no chance to get lost, Thesis, R. U. Utrecht, 1983. MR 765860 (86j:58069)
  • [13] C. Preston, Iterates of maps on an interval., Lecture Notes in Math., vol. 999, Springer-Verlag, 1983. MR 706078 (85c:58058)
  • [14] A. N. Sharkovsky, Coexistence of the cycles of a continuous mapping of the line into itself, Ukrain. Math. Zh. 16 (1964), 61-71. MR 0159905 (28:3121)
  • [15] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260-267. MR 0494306 (58:13206)
  • [16] S. Smale and R. F. Williams, The qualitative analysis of a difference equation of population growth, J. Math. Biol. 3 (1976), 1-4. MR 0414147 (54:2251)
  • [17] P. Stefan, A theorem of Sharkovsky on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237-248. MR 0445556 (56:3894)
  • [18] P. D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (178), 99-105. MR 498731 (80h:58043)
  • [19] D. Whitley, Discrete dynamical systems in dimensions one and two, Bull. London Math. Soc. 15 (1983), 177-217. MR 697119 (84h:58079)

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Additional Information

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

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