Contributions to the geometric and ergodic theory of conservative flows (Q2874002)

The aim of the paper under review is twofold: first the authors obtain \textit{J. Bochi} and \textit{M. Viana}'s theorem [Ann. Math. (2) 161, No. 3, 1423--1485 (2005; Zbl 1101.37039)] for the continuous time case and second they prove the flow counterpart of a result by \textit{J. Bochi} et al. [C. R., Math., Acad. Sci. Paris 342, No. 10, 763--766 (2006; Zbl 1097.37010)].NEWLINENEWLINEIndeed, they show that, from a \(C^1\)-generic perspective, only two opposing situations can occur almost surely: either zero Lyapunov exponents or uniform projective hyperbolicity (a.k.a. dominated splitting, exponentially separated). Let us state the first result.NEWLINENEWLINETheorem 1. There exists a \(C^1\)-residual set \(\mathcal{E} \subset \mathfrak{X}^{1}_\mu(M)\) such that if \(X \in \mathcal{E}\) then there exist two \(X^t\)-invariant subsets of \(M\), \(\mathcal{Z}\) and \(\mathcal{D}\), such that their union has full measure and: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] if \(p \in \mathcal{Z}\) then all the Lyapunov exponents associated to \(p\) are zero; \item[{\(\bullet\)}] if \(p \in \mathcal{D}\) then its orbit admits a dominated splitting for the linear Poincaré flow. NEWLINENEWLINE\end{itemize}} Then, the authors use the previous theorem in order to obtain the following result which assures non-zero Lyapunov exponents and prevalence of dominated splitting for stably ergodic flows.NEWLINENEWLINETheorem 2. There exists a \(C^1\)-open and dense set \(\mathcal{U}\) inside the space of \(C^1\)-stably ergodic flows in \(\mathfrak{X}^{1+\alpha}_\mu(M)\), \(\alpha>0\), such that if \(X \in \mathcal{U}\) then \(X^t\) is a non-uniformly hyperbolic flow and \(X\) admits a dominated splitting for the linear Poincaré flow that separates the spaces corresponding to positive and negative Lyapunov exponents.NEWLINENEWLINEThe approach followed for the proofs relies on the study of the action of the tangent flow on the normal sub-bundle (linear Poincaré flow) and the construction of perturbation results in order to perform the required changes on the normal dynamics. Related to this purpose, an improved version of the Pasting Lemma [\textit{A. Arbieto} and \textit{C. Matheus}, Ergodic Theory Dyn. Syst. 27, No. 5, 1399--1417 (2007; Zbl 1142.37025)] is proved (Lemma 5.2), which is then used to obtain several elaborated perturbations of a volume-preserving flow. These perturbations intend to rotate the vectors in the normal sub-bundles with the objective of interchanging the rate of expansion/contraction of the linear Poincaré flow.