General form of fixed point indices of an iterated C¹ map and infiniteness of minimal periods

Abstract

Let f be a smooth self-map of a compact manifold and be a family of compact subsets of periodic points of f. Under some natural condition on the family we find the form of the sequence of indices of iterations , which generalizes the classical theorem of Chow, Mallet-Paret and Yorke. We apply this knowledge to study the structure of periodic points of f. In particular, we show that a map f with unbounded sequence of Lefschetz numbers of iterations , which satisfies some assumption put on derivatives at periodic points, has an infinite number of minimal periods.

Keywords:

• fixed point index
• periodic points
• iterations
• C ¹ maps

AMS Subject Classifications:

• Primary 37C25
• Secondary 37C05

Acknowledgement

The research was supported by KBN grant No. 1 P03A 03929.