Quasi-shadowing for partially hyperbolic dynamics on Banach spaces (Q2657701)

The use of shadowing in dynamical systems has been very effective in problems that involve topological stability. While uniformly hyperbolic systems in discrete and continuous time have very useful shadowing properties, the situation is more complicated with partially hyperbolic systems. One cannot expect to recover full shadowing behavior for partially hyperbolic systems, even those that are robustly transitive. The authors' main result is that it is possible to recover a weaker property that they call quasi-shadowing. The setting is as follows: \(M = (M,d)\) is a metric space, and \(F_n(y_n)\) for \(n \in \mathbb{Z}\) is a sequence of maps on \(M\). If \(\delta > 0\) and \(y_n\) is a sequence of points in \(M\) satisfying \(d(y_{n+1}, F_n (y_n) )< \delta\) for every \(n \in \mathbb{Z}\), then a sequence of points \((y_n)\) is called a \(\delta\)-pseudotrajectory for the nonautonomous dynamics arising from \(x_{n+1} = F_n (x_n)\). The latter system is said to have the shadowing property if for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that for any \(\delta\)-pseudotrajectory \((y_n)\) of the system there is a sequence of points \((x_n)\) in \(M\) satisfying \(x_{n+1} = F(x_n)\) such that \(d(x_n, y_n) < \epsilon\) for every \(n \in \mathbb{Z}\). A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points \((x_n)\) such that \(x_{n+1}\) is obtained from the image of \(x_n\) by moving it by a small factor in the central direction. The authors move from the finite-dimensional setting to an infinite-dimensional Banach space and Banach sequence space. They start with nonautonomous dynamical systems that are not necessarily invertible on an infinite dimensional space, thus beginning with linear dynamics of the form \(x_{m+1} = A_m x_m\) for integers \(m\), where the sequence (\(A_m\)) admits a partial dichotomy. They prove that a small nonlinear perturbation of this system has the quasi-shadowing property. The proof of this relies on finding the fixed point of a certain contraction mapping, and then using that fixed point to construct the quasi-trajectory. The authors also state and prove a continuous-time version of the same result.