Topological stability for conservative systems (Q2430946)

This article deals with flows on \(n\)-dimensional (\(n \geq 3\)) closed and connected \(C^\infty\) Riemannian manifolds \(M\) endowed with a volume-form \(\omega\). A vector field \(X: M \to M\) is divergence-free and an associated with \(X\) flow is incompressible if \(\nabla \cdot X = 0\) (or equivalently, the measure \(\mu\) associated to \(\omega\) is invariant for associated flow \(X^t: M \to M\), \(t \in {\mathbb R}\)). A vector field \(X: M \to M\) is topologically stable if for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that for any \(Y\) \(\delta\)-close to \(X\), there exists a semiconjugacy from \(Y\) to \(X\), i.e. there exists \(h: M \to M\) and \(\tau: M \times {\mathbb R} \to {\mathbb R}\) satisfying the following properties (a) for any \(x \in M\), \(\tau_x: {\mathbb R} \to {\mathbb R}\) is an orientation preserving homeomorphism with \(\tau(x,0) = 0\) and (b) for all \(x \in M\) and \(t \in {\mathbb R}\) \(h(Y^t(x)) = X^{\tau(x,t)}(h(x))\) holds, and such that \(d(h(x),x) < \varepsilon\) for all \(x \in M\). A vector field \(X: M \to M\) is called Anosov if its tangent bundle \(TM\) splits into three continuous \(DX^t\)-invariant nontrivial subbundles \(E^0 \oplus E^1 \oplus E^2\) where \(E^0\) is the flow direction, the subbundle \(E^2\) is uniformly contracted and the subbundle \(E^1\) is uniformly expanded for \(t > 0\). The authors prove that each vector field \(X \in \text{int}_{C^1}({\mathcal TS}_\mu(M))\) is Anosov; here \({\mathcal TS}_\mu(M)\) is the set of topological stable incompressible flows. An analogous result for discrete-time case is also proved.