2018/06/30 by Vitor Araujo
Mathematics · #math.DS #math.CA #msc:37D25 #msc:37D30 #msc:37D20
published as Osaka J. Math. 57 (2020), no. 4, 751--788 · 37 pages; 10 figures; presentation improved after referee's report and correction of imprecise statements of lemmas 3.3, 3.4 and 3.5, and small adjustements in the proofs. To appear in Osaka J Math 2020
arxiv created 2020/05/11 · arxiv updated 2021/07/27
Let f:M→ M be a C1 map of a compact manifold M, with dimension at least 2, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume that Df is never the null map at any point (in particular, we need no extra smoothness assumption on Df), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a C1 diffeomorphism is itself a periodic repeller (source). Analogously for a C1 open and dense subset of vector field on finite dimensional manifolds: for a flow ϕt generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincaré Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the C1+ diffeomorphism setting to C1 endomorphisms and C1 flows. Some ergodic theoretical consequences are discussed. The proofs use versions of Pliss' Lemma for maps and flows translated as (reverse) hyperbolic times, and a result ensuring that certain subadditive cocycles over vector fields are in fact additive.