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 $\pm s$, $\sqrt 2 \leqslant s \leqslant 2$. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters $s$, 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.