The local \(C^1\)-density of stable ergodicity (Q379447)

Let \(M\) be a smooth compact manifold with \(\text{dim}M\geq 3\). Let \({\mathcal P}\) be the set of \(C^1\) partially hyperbolic diffeomorphisms \(f\) that preserve a smooth volume measure on \(M\) and have the following property: \(f\) has a \(C^1\) neighborhood \(U\) such that any \(g\in U\) has two ergodic measures \(\mu_1\) and \(\mu_2\) such that all center Lyapunov exponents are nonnegative for \(\mu_1\) and nonpositive \(\mu_2\). The author shows that there exists a \(C^1\) dense subset \(D\) of \({\mathcal P}\) such that any \(f\in D\) is stably ergodic.