SINKS AND SOURCES FOR C1 DYNAMICS WHOSE LYAPUNOV EXPONENTS HAVE CONSTANT SIGN

Let f : M → M be a C^1 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 nor the existence of a invariant probability measure), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a C^1 diffeomorphism is itself a periodic repeller (source). Analogously for a C^1 open and dense subset of vector field on finite dimensional manifolds: for a flow φ_t 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 C^ diffeomorphism setting to C^1 endomorphisms and C^1 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 C^1 vector fields are in fact additive