Periodic points on tori

Benjamin Rigler Halpern

Vol. 83 (1979), No. 1, 117–133

DOI: 10.2140/pjm.1979.83.117

Abstract

We prove the following theorem.

Theorem 1. Given a continuous map f : Tⁿ → Tⁿ of the n-dimensional torus into itself. Each map homotopic to f has an infinite number of periodic points if and only if the Lefschetz numbers of the iterates L(f^m), m = 1,2,⋯, are unbounded.

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 54H25, 58F20

Milestones

Received: 15 February 1978

Revised: 4 December 1978

Published: 1 July 1979