Pseudo-Orbit Shadowing in the Family of Tent Maps

Abstract

We study the family of tent maps-continuous, unimodal, piecewise linear maps of the interval with slopes s, \/2 < s < 2. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters s, although they fail to have the shadowing property for an uncountable, dense set of parameters.We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.