Finding periodic points of a map by use of a K-adic expansion

Discrete and Continuous Dynamical Systems

Volumn 5, Issue 3, 1999, Pages 495-514

  Marzantowicz, Wacław ^a,b   Przygodzki, Piotr Maciej ^a  

^a UNIVERSITY OF GDA SK   (Poland)

^b ADAM MICKIEWICZ UNIVERSITY   (Poland)

Author keywords

Arithmetic function; Fixed point index; Lefschetz number; Minimal period; Periodic point

Indexed keywords

EID: 0033463475     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.1999.5.495     Document Type: Article

References (26)

1. T. Apostol, "Modular Functions and Dirichlet Series in Number Theory", Graduate Texts in Mathematics, Springer-Verlag, 2nd Edition, 1990.
2. I.K. Babienko, S.A. Bogatyi, Behavior of the index of periodic points under iterations of a mapping, Izv.-Akad.-Nauk-SSSR-Ser.-Mat., 55 no. 1, (1991), 3-31 (in Russian). English translation: Math.-USSR-Izv. [Mathematics-of-the-USSR-Izvestiya], 38 no. 1, (1992), 1-26.
3. I.K. Babienko, S.A. Bogatyi, Behavior of the index of periodic points under iterations of a mapping, Izv.-Akad.-Nauk-SSSR-Ser.-Mat., 55 no. 1, (1991), 3-31 (in Russian). English translation: Math.-USSR-Izv. [Mathematics-of-the-USSR-Izvestiya], 38 no. 1, (1992), 1-26.
4. C. Bowszyc, On the Euler characteristic of a map and the existence of periodic points, Bull. de L'Acad. Polon. des Sci., Vol. XVII, No. 6 (1969), 367-372.
5. R.F Brown, "The Lefschetz Fixed Point Theorem", Scott, Foresman and Comp., Glenview Illinois, 1972.
6. N. Bourbaki, "Elements de Mathematique", XI, Livre II, Algebre, Hermman, Paris 1949.
7. J. Casasayas, J. Llibre and A. Nunes, Periodic orbits of transversal maps, Math. Proc. Camb. Phil. Soc. 118, (1995), 161-181.
8. K. Chandrasekharan, "Introduction to Analytic Number Theory", Springer-Verlag, Berlin, Heidelberg-New York, 1968.
9. S.N. Chow, J. Mallet-Paret, J. A. Yorke, A bifurcation invariant: Degenerate orbits treated as clusters of simple orbits, Geometric dynamics: the Proceedings of a Dynamics Meetings at IMPA in Rio de Janeiro (August 1981), Lect. Notes Math. vol. 1007 (1983), 109-131.
10. A. Dold, "Lectures on Algebraic Topology", Springer-Verlag, 1972.
11. A. Dold, Fixed point indices of iterated maps, Inventiones Math., 74 (1985), 419-435.
12. A. Dold, Combinatorial and geometric fixed point point theory, Riv. Math. Univ. Parma, (4) 10* (1984), 23-32.
13. A. Dold, Period doubling and fixed point indices, notes, Heidelberg, August 1986.
14. J. Dugundji, A. Granas, "Fixed Point Theory", Monografie Matematyczne, PWN, Warszawa, 1982.
15. L. Erbe, K. Gȩba and W. Krawcewicz, Equivariant fixed point index and the period-doubling cascades, Can. Journ. of Math., 43 (4) (1991), 738-747.
16. J. Franks, Period doubling and the Lefschetz formula, Trans AMS, 287 (1985), 275-283.
17. F.B. Fuller, The existence of periodic points, Ann. of Math., 57 (1953), 229-230.
18. K. Komiya, Fixed point indices of equivariant maps and Möbius inversion, Inventiones Math., 91 (1988), 129-135.
19. M.A. Krasnosielski, P. P. Zabrejko, The rotation of vector fields with superpositions and iterations of operators, Vestnik Jarosl. Univ. 12 (1975), 23-37 (in Russian).
20. T. Matsuoka, The number of periodic points of smooth maps, Ergod. Th. & Dynam. Sys., 9 (1989), 153-163.
21. W. Marzantowicz, Finding Periodic points of smooth mapping using Lefschetz Number of iterations, Izviestya Vuzov RRF, vol. 417, No 2, (1997), 86-96.
22. P. Seidel, Trasversalität für periodische Punkte differenziebarer Abbildungen, Mathematisches Institut, Universität Heidelberg, Forschergruppe: Topologie und nichtkommutative Geometrie, Preprint Nr 97, July 1994.
23. J. P. Serre, "Representations of Finite Groups", Springer-Verlag, 1987.
24. M. Shub, Dynamical systems, filtrations and entropy, Bull. AMS, 80 (1974), 21-24.
25. M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology, 13 (1974), 189-191.
26. H. Steinlein, Satz über den Leray-Schauderschen Afabildungsgrad Math. Zeit., 126 (1972), 176-208