Sinks and sources for C1 dynamics whose Lyapunov exponents have constant sign

TL;DR

This paper characterizes the behavior of trajectories with constant sign Lyapunov exponents in $C^1$ dynamical systems, showing they converge to sinks or repellers and extending non-uniform hyperbolic theory to less smooth contexts.

Contribution

It extends Pesin's theory to $C^1$ systems, establishing sinks and sources for trajectories with uniform Lyapunov exponents without requiring high smoothness.

Findings

Trajectories with negative Lyapunov exponents converge to periodic sinks.

Trajectories with positive Lyapunov exponents are periodic repellers.

Results apply to $C^1$ endomorphisms and flows, broadening hyperbolic theory.

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 $Df$), 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 Poincar\'e 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 vector fields are in fact additive.