Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

Abstract

A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n{)}_{n\in \mathbb{Z}}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction.

Minor changes. Accepted for publication in Journal of Mathematical Analysis and Applications. arXiv admin note: text overlap with arXiv:1905.08251