The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

ZETA FUNCTIONS FOR ONE-DIMENSIONAL GENERALIZED SOLENOIDS

  • Yi, In-Hyeop (Department of Mathematics Education, Ewha Womans University)
  • Received : 20110100
  • Accepted : 2011.02.26
  • Published : 2011.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

We compute zeta functions of 1-solenoids. When our 1-solenoid is nonorientable, we compute Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid and its orientable double cover explicitly in terms of adjacency matrices and branch points. And we show that Artin-Mazur zeta function of orientable double cover is a rational function and a quotient of Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid.

Keywords

References

  1. M. Artin & B. Mazur: On periodic points. Ann. of Math. 81 (1965), 82-99. https://doi.org/10.2307/1970384
  2. A. Fel'shtyn: Dynamical zeta functions, Nielsen theory and Reidemeister torsion. Mem. Amer. Math. Soc. 147 (2000), no. 699.
  3. R. Fokkink: The structure of trajectories. Ph.D. Thesis, Technische Universiteit te Delft, 1991.
  4. J. Franks: A reduced zeta functions for diffeomorphisms. Amer. J. Math. 100 (1978), 217-243. https://doi.org/10.2307/2373849
  5. D. Ruelle: Dynamical zeta functions for piecewise monotone maps of the interval. CRM Monograph Series 4, Amer. Math. Society, 1994.
  6. S. Smale: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747-817. https://doi.org/10.1090/S0002-9904-1967-11798-1
  7. R.F. Williams: One-dimensional non-wandering sets. Topology 6 (1967), 473-487. https://doi.org/10.1016/0040-9383(67)90005-5
  8. R.F. Williams: Classification of 1-dimensional attractors. Proc. Symp. Pure Math. 14 (1970), 341-361.
  9. R.F. Williams: Expanding attractors. IHES Publ. Math. 43 (1974), 169-203. https://doi.org/10.1007/BF02684369
  10. I. Yi: Canonical symbolic dynamics for one-dimensional generalized solenoids. Trans. Amer. Math. Soc. 353 (2001), 3741-3767. https://doi.org/10.1090/S0002-9947-01-02710-6
  11. I. Yi: Ordered group invariants for nonorientable one-dimensional generalized solenoids. Proc. Amer. Math. Soc. 131 (2003), 1273-1282. https://doi.org/10.1090/S0002-9939-02-06794-1
  12. A.Yu. Zhirov: Hyperbolic attractors of diffeomorphisms of orientable surfaces. Russian Acad. Sci. Sb. Math. 82 (1995), 135-174.