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July  1999, 5(3): 495-514. doi: 10.3934/dcds.1999.5.495

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

1. 

Institute of Mathematics, University of Gdańsk, Gdańsk, Poland, Poland

Received  May 1995 Revised  March 1999 Published  May 1999

In this paper we develop a theory of k-adic expansion of an integral aritmethic function. Applying this formal language to Lefschetz numbers, or fixed point indices, of iterations of a given map we reformulate or reprove earlier results of Babienko-Bogatyj, Bowszyc, Chow-Mallet-Paret and Franks. Also we give a new characterization of a sequence of Lefschetz numbers of iterations of a map $f$: For a smooth transversal map we get more refined version of Matsuoka theorem on parity of number of orbits of a transversal map. Finally, for any $C^1$-map we show the existence of infinitely many prime periods provided the sequence of Lefschetz numbers of iterations is unbounded.
Citation: Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495
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