Some results on perturbations of Lyapunov exponents (Q1760088)

This paper studies two properties of the Lyapunov exponents of a diffeomorphism under small perturbations: one is when the zero Lyapunov exponents can be removed and the other is when all the Lyapunov exponents can be distinguished. The first result shows that for any volume preserving partially hyperbolic diffeomorphism \(f\) and any neighborhood \(\mathcal{U}\) of \(f\), there is a volume preserving partially hyperbolic diffeomorphism \(g\) with non-zero integrated Lyapunov exponents \(\int_M \lambda_j(g,x)\,d\omega(x)\neq 0\), \(1\leq j \leq d\), where \(\omega\) is a smooth volume form on a \(d\)-dimensional compact Riemannian manifold \(M\). The second part of this paper contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between the simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among conservative diffeomorphisms far from tangencies, the authors prove that the ergodic ones form a residual subset.