Sinks and sources for $C^1$ dynamics whose Lyapunov exponents have constant sign

Abstract

Let $f:M\to M$ 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 $\phi_t$ 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 $C^{1+}$ 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.