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\(C^1\)-generic sectional axiom A flows have only trivial symmetries. (English) Zbl 1440.37038

Let \(X\in \mathfrak{X}^1(M)\) be a \(C^1\) vector field on an \(n\)-dimensional manifold \(M\). A compact invariant set \(\Lambda\) is said to be (uniformly) hyperbolic if there exists a three way splitting \(T_{x}M = E^s_{x} \oplus \langle X(x) \rangle \oplus E^u_x\) of the tangent space over \(x \in \Lambda\) that is invariant under the flow \(X_t\), \(t\in \mathbb{R}\) generated by \(X\) such that \(DX_t|_{E^s_x}\) is uniformly contracting and \(DX_t|_{E^u_x}\) is uniformly expanding. This definition does not work if the vector field \(X\) has a singularity on \(\Lambda\), that is, \(X(\sigma) = 0\) for some \(\sigma \in \Lambda\). The notion of singular hyperbolicity for three-dimensional flows was introduced in [C. A. Morales et al., Ann. Math. (2) 160, No. 2, 375–432 (2004; Zbl 1071.37022)], and its higher-dimensional version of sectional hyperbolicity in [R. Metzger and C. Morales, Ergodic Theory Dyn. Syst. 28, No. 5, 1587–1597 (2008; Zbl 1165.37010)].
The authors prove that \(C^1\)-generic Axiom A flows have trivial centralizer and that \(C^1\)-generic sectional Axiom A flows on a three-dimensional manifold have trivial centralizer. Similar results for diffeomorphisms have been obtained in [C. Bonatti et al., J. Mod. Dyn. 2, No. 2, 359–373 (2008; Zbl 1149.37017); Publ. Math., Inst. Hautes Étud. Sci. 109, 185–244 (2009; Zbl 1177.37025)].

MSC:

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C10 Dynamics induced by flows and semiflows
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37D05 Dynamical systems with hyperbolic orbits and sets
37D10 Invariant manifold theory for dynamical systems

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