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

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

MathSciNet review:
911091

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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.

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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