Persistence of Semi-Trajectories

We consider diffeomorphisms f of a smooth compact riemannian mainfold M and its suspension flow $(M, \phi)$. Assuming some regularity of the stable (unstable) sets at the points $f^n(x), \phi_{t}(x), n \geq 0, t \geq 0$ we prove the persistence in the future of {f ⁿ(x), n ≥ 0} or $\{\phi_{t}(x), t \geq 0\}$, i.e., that C ⁰ small perturbations g of f have a semi-trajectory that closely shadows {f ⁿ(x), n ≥ 0} and that the suspension of g has also a semi-trajectory that closely shadows $\{\phi_{t}(x), t \geq 0\}$. In case x belongs to a minimal set of f we show that the assumptions concerning the regularity of stable and unstable sets could be reduced to a neighbourhood of x.