On \(C^0\) genericity of various shadowing properties
Let \((M,d)\) be a compact metric space and let \(f:M\rightarrow M\) be a homeomorphism (a discrete dynamical system on \(M\)). A sequence \(\{ y_n \}_{n\in {\mathbb Z}}\subset M\) is called a \(\delta\)-pseudo-orbit (\(\delta\geq 0\)) of \(f\) if \(d(f(x_n )\), \(x_{n+1})\leq \delta\), \(\forall n \in {\mathbb Z}\). A system \(f\) is said to have the periodic shadowing property if \(\forall \varepsilon >0\quad \exists \delta >0\) satisfying the condition: given a periodic \(\delta\)-pseudo-orbit \(y=\{ y_n \}\) there exists an orbit \(x=\{ x_n \}\) such that \(x\) \(\varepsilon \)-traces \(y\), i.e., \(d(x_n ,y_n )\leq \varepsilon \quad \forall n\in {\mathbb Z}\). Let \(M^{\mathbb Z}\) be the compact of two-sided sequences \(x=\{ x_n \}_{n\in{\mathbb Z}}\subset M\), endowed with the product topology. The \(\delta\)-method for \(f\) is a mapping \(\phi :M\rightarrow M^{\mathbb Z}\) such that \(\forall x\in M\) the sequence \(\phi (x)\) is a \(\delta\)-pseudo-orbit of \(f\). A system \(f\) is said to have the inverse shadowing property with respect to a class \(\mathcal T\) if \(\forall \varepsilon >0\) \(\exists\delta >0\), such that given a \(\delta\)-method \(\phi\in{\mathcal T}, \forall x=\{x_n \}\exists y\in M\) such that \(\delta\)-pseudo-orbit \(\phi (y)\) \(\varepsilon\)-traces \(x\). Denote \({\mathcal T}_C =\{\phi :M\rightarrow M^{\mathbb Z}\mid \phi\) is a continuous \(\delta\)-method for some \(\delta >0\}\), \({\mathcal H}(M)\) is the set of all homeomorphisms of \(M\). Main results of the paper are: 1. If the space \(M\) is a closed smooth manifold, then periodic shadowing is a generic property in \({\mathcal H}(M)\). 2. If the space \(M\) is a closed manifold of the dimension \(\leq 3\), then \({\mathcal T}_C\)-inverse shadowing is a generic property in \({\mathcal H}(M)\).
scientific article; zbMATH DE number 2155035