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

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  • 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.
    Mathematics Subject Classification: 55M20, 58F20, 54H25, 58C30.

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