Generic homeomorphisms do not have the Lipschitz shadowing property (Q1569793)

The subject of the paper are discrete dynamical systems (homeomorphisms) on a smooth closed manifold \(M\). The considered problem is shadowing of pseudotrajectories, i.e. finding trajectories that are close in a certain sense to its approximate \(\delta\)-trajectories. A system \(\varphi\) has the Lipshitz shadowing property (with constants \(L, d>0)\) if, for every \(\delta\in (0,d)\) and an arbitrary \(\delta\)-trajectory \(\{ x_k \}\), there exists a point \(x\in M\) for which \(\text{ dist} (\varphi^k(x),x_k)<L\delta, k\in Z\), where \(\text{dist}\) is a Riemann metric on a manifold. The main result states that generic homeomorphisms do not have the Lipshitz shadowing property. The appropriate space \(Z(M)\) of dynamical systems consists of homeomorphisms on \(M\) with a topology induced by the metric \(\rho(\varphi,\psi) = \max \{ \text{dist} (\varphi(x),\psi(x))\), \(\text{dist} (\varphi^{-1}(x),\psi^{-1}(x)) \}\).