Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sinks and sources for C1 dynamics whose Lyapunov exponents have constant
  sign   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.

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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas