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

Full-text 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 (one-sided) 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), no. 4, 687–691. MR**0475982****[2]**M. Artin and B. Mazur,*On periodic points*, Ann. of Math. (2)**81**(1965), 82–99. MR**0176482****[3]**Louis Block,*Simple periodic orbits of mappings of the interval*, Trans. Amer. Math. Soc.**254**(1979), 391–398. MR**539925**, 10.1090/S0002-9947-1979-0539925-9**[4]**Rufus Bowen,*Markov partitions for Axiom 𝐴 diffeomorphisms*, Amer. J. Math.**92**(1970), 725–747. MR**0277003****[5]**Pierre Collet and Jean-Pierre Eckmann,*Iterated maps on the interval as dynamical systems*, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR**613981****[6]**John Guckenheimer,*On the bifurcation of maps of the interval*, Invent. Math.**39**(1977), no. 2, 165–178. MR**0438399****[7]**Tien Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**0385028****[8]**J. Milnor and W. Thurston,*On iterated maps of the interval*. I*and*II, Mimeographed, Princeton Univ., 1977.**[9]**Michał Misiurewicz,*Structure of mappings of an interval with zero entropy*, Inst. Hautes Études Sci. Publ. Math.**53**(1981), 5–16. MR**623532****[10]**Michał Misiurewicz,*Absolutely continuous measures for certain maps of an interval*, Inst. Hautes Études Sci. Publ. Math.**53**(1981), 17–51. MR**623533****[11]**Zbigniew Nitecki,*Topological dynamics on the interval*, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR**670074****[12]**Helena Engelina Nusse,*Chaos, yet no chance to get lost*, Drukkerij Elinkwijk B. V., Utrecht, 1983. Dissertation, Rijksuniversiteit te Utrecht, Utrecht, 1983. MR**765860****[13]**Chris Preston,*Iterates of maps on an interval*, Lecture Notes in Mathematics, vol. 999, Springer-Verlag, Berlin, 1983. MR**706078****[14]**O. M. Šarkovs′kiĭ,*Co-existence of cycles of a continuous mapping of the line into itself*, Ukrain. Mat. Z.**16**(1964), 61–71 (Russian, with English summary). MR**0159905****[15]**David Singer,*Stable orbits and bifurcation of maps of the interval*, SIAM J. Appl. Math.**35**(1978), no. 2, 260–267. MR**0494306****[16]**S. Smale and R. F. Williams,*The qualitative analysis of a difference equation of population growth*, J. Math. Biol.**3**(1976), no. 1, 1–4. MR**0414147****[17]**P. Štefan,*A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line*, Comm. Math. Phys.**54**(1977), no. 3, 237–248. MR**0445556****[18]**Philip D. Straffin Jr.,*Periodic points of continuous functions*, Math. Mag.**51**(1978), no. 2, 99–105. MR**498731**, 10.2307/2690145**[19]**David Whitley,*Discrete dynamical systems in dimensions one and two*, Bull. London Math. Soc.**15**(1983), no. 3, 177–217. MR**697119**, 10.1112/blms/15.3.177

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:
https://doi.org/10.1090/S0002-9947-1987-0911091-1

Keywords:
Iteration of mappings,
periodic points,
semigroup of chaotic mappings,
zeta function

Article copyright:
© Copyright 1987
American Mathematical Society