Shadowing is generic
Let \(M\) be a smooth compact manifold without boundary. Let \(r\) be a Riemannian metric on \(M\). Consider the space \(Z(M)\) of homeomorphisms \(M\to M\) equipped with the \(C^0\) topology induced by the metric \(\rho(\varphi,\psi)=\max_{x\in M} r(\varphi(x),\psi(y))\). A sequence \(x_k\in M\), \(k\in Z\), is called a \(d\)-pseudotrajectory for \(\varphi\in Z(M)\) if \(r(x_{k+1},\varphi(x_k))<d\), \(k\in Z\). We say that \(\varphi\) has the shadowing property if for any \(\varepsilon>0\) there is \(d>0\) such that for every \(d\)-pseudotrajectory \(\{x_k\}\) there exists a point \(x\in M\) such that \(r(x_k,\varphi^k(x))<\varepsilon\), \(k\in Z\). The shadowing property is a key tool in the studies of chaotic maps. It is shown here that a generic homeomorphism \(\varphi\in Z(M)\) has the shadowing property. Previously, this fact was only proved when dim\(M\leq 3\).
scientific article; zbMATH DE number 1367547