Partial hyperbolicity and robust transitivity (Q1588919)

Let \(M\) denote a three-dimensional boundaryless compact manifold and \(\text{Diff}(M)\) the space of \(C^1\)-diffeomorphism defined on \(M\) endowed with the usual \(C^1\)-topology. The maximal invariant set of \(\varphi\) in an open set \(U\), denoted by \(\Lambda_\varphi(U)\), is the set of points whose whole orbit is contained in \(U\). This paper is devoted to the hyperbolicity (uniform partial and strong partial) of \(\Lambda_4(U)\) derived from its robust transitivity. In the case \(U=M\), \(\varphi\) is robustly transitive.