Chaotic maps with rational zeta function
Author:
H. E. Nusse
Journal:
Trans. Amer. Math. Soc. 304 (1987), 705719
MSC:
Primary 58F13; Secondary 58F14, 58F20
MathSciNet review:
911091
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Fix a nontrivial interval and let be a chaotic mapping. We denote by the set of points whose orbits do not converge to a (onesided) asymptotically stable periodic orbit of or to a subset of the absorbing boundary of 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) is compact, is expanding on . Then we can associate a matrix with entries either zero or one to the mapping such that the number of periodic points for with period is equal to the trace of the matrix ; furthermore the zeta function of is rational having the eigenvalues of as poles. B. We assume that such that: (1) the Schwarzian derivative of is negative, and (2) the closure of is compact and for all in the closure of . Then we obtain the same result as in A.
 [1]
D. J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34 (1978), 687691. MR 0475982 (57:15563)
 [2]
M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 8299. MR 0176482 (31:754)
 [3]
L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391398. MR 539925 (80m:58031)
 [4]
R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725747. 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), 165178. MR 0438399 (55:11312)
 [7]
T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985992. 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), 516. MR 623532 (83j:58071)
 [10]
, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 1751. 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. 173. 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, SpringerVerlag, 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), 6171. MR 0159905 (28:3121)
 [15]
D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260267. 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), 14. 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), 237248. MR 0445556 (56:3894)
 [18]
P. D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (178), 99105. MR 498731 (80h:58043)
 [19]
D. Whitley, Discrete dynamical systems in dimensions one and two, Bull. London Math. Soc. 15 (1983), 177217. MR 697119 (84h:58079)
 [1]
 D. J. Allwright, Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34 (1978), 687691. MR 0475982 (57:15563)
 [2]
 M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 8299. MR 0176482 (31:754)
 [3]
 L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc. 254 (1979), 391398. MR 539925 (80m:58031)
 [4]
 R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725747. 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), 165178. MR 0438399 (55:11312)
 [7]
 T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985992. 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), 516. MR 623532 (83j:58071)
 [10]
 , Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 1751. 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. 173. 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, SpringerVerlag, 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), 6171. MR 0159905 (28:3121)
 [15]
 D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260267. 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), 14. 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), 237248. MR 0445556 (56:3894)
 [18]
 P. D. Straffin, Periodic points of continuous functions, Math. Mag. 51 (178), 99105. MR 498731 (80h:58043)
 [19]
 D. Whitley, Discrete dynamical systems in dimensions one and two, Bull. London Math. Soc. 15 (1983), 177217. MR 697119 (84h:58079)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
58F13,
58F14,
58F20
Retrieve articles in all journals
with MSC:
58F13,
58F14,
58F20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198709110911
PII:
S 00029947(1987)09110911
Keywords:
Iteration of mappings,
periodic points,
semigroup of chaotic mappings,
zeta function
Article copyright:
© Copyright 1987
American Mathematical Society
