For any C1 perturbation of a vector field with a sectional hyperbolic set, the number of homoclinic classes inside a fixed neighborhood is uniformly bounded.
Morse-Smale systems and horseshoes for three dimensional singular flows
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abstract
We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.
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math.DS 1years
2019 1verdicts
REJECT 1representative citing papers
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Finiteness of homoclinic classes on sectional hyperbolic sets
For any C1 perturbation of a vector field with a sectional hyperbolic set, the number of homoclinic classes inside a fixed neighborhood is uniformly bounded.