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 \pm s , 2 ⩽ s ⩽ 2 \sqrt 2 \leqslant s \leqslant 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 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.