\(C^ 0\) generic properties of stable and unstable sets of diffeomorphisms. (Q2782716)

Let \(\mathcal{H}om(M)\) be the space of homeomorphisms on an \(n\) Riemanian manifold \((M,d)\) endowed with the \(C^{0}\) topology induced by the complete metric NEWLINE\[NEWLINE d_{0}(f,g)=\max_{x\in M}\left\{ d\left( f(x),g(x)\right) ;d\left( f^{-1}(x),g^{-1}(x)\right) \right\} . NEWLINE\]NEWLINE For a given an \(\varepsilon>0,\) define the \(\varepsilon\)-local stable set of \(x\in M\) with respect to \(f\in\mathcal{H}om(M)\) by NEWLINE\[NEWLINE S_{\varepsilon}^{f}(x)=\left\{ y\in M:d\left( f^{n}(x),f^{n}(y)\right) \leq\varepsilon\text{ for all }n\geq0\right\} NEWLINE\]NEWLINE and the \(\varepsilon\)-local unstable set of \(x\in M\) with respect to \(f\in\mathcal{H}om(M)\) by NEWLINE\[NEWLINE U_{\varepsilon}^{f}(x)=\left\{ y\in M:d\left( f^{n}(x),f^{n}(y)\right) \leq\varepsilon \text{ for all }n\leq0\right\}. NEWLINE\]NEWLINE The paper is concerned with the description of the sets \(S_{\varepsilon} ^{f}(x)\) and \(U_{\varepsilon}^{f}(x)\) when \(x\) belongs to the nonwandering set \(\Omega(f)\) of \(f.\) In particular, the author studies the question related to separation properties of \(S_{\varepsilon}^{f}(x)\) and \(U_{\varepsilon} ^{f}(x).\) The main result states that on a \(C^{0}\) residual set of diffeomorphisms \(f,\) all \(\varepsilon\)-local stable and unstable sets \(S_{\varepsilon}^{f}(x)\) and \(U_{\varepsilon}^{f}(x)\) of points \(x\) in the nonwandering set \(\Omega(f)\) of \(f\) contain nontrivial compact connected pieces which have, respectively, \(s\)- and \(u\)-separation properties with respect to some open ball \(B_{\lambda}(x),\) where \(s\) and \(u\) are complementary dimensions. Furthermore, they are placed in \(B_{\lambda}(x)\) in a way that resembles transversality, imitating what occurs in dynamical models which are better explored.