Ergodic Theory and Dynamical Systems

Research Article

Homoclinic and non-wandering points for maps of the circle

Louis Blocka1, Ethan Covena2, Irene Mulveya3 and Zbigniew Niteckia4

a1 Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

a2 Department of Mathematics, Wesleyan University, Middleton, CT 06457, USA

a3 Department of Mathematics, Swarthmore College, Swarthmore, PA 19081, USA

a4 Department of Mathematics, Tufts University, Medford, MA 02155, USA

Abstract

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.

(Received January 04 1983)

(Revised May 15 1983)