TL;DR
This paper proves that for generic $C^1$ diffeomorphisms, any expansive homoclinic class must be hyperbolic, establishing a significant link between expansiveness and hyperbolicity in dynamical systems.
Contribution
It demonstrates that expansive homoclinic classes are necessarily hyperbolic for $C^1$ generic diffeomorphisms, a novel result connecting these concepts.
Findings
Expansive homoclinic classes are hyperbolic for generic $C^1$ diffeomorphisms.
Establishes a fundamental link between expansiveness and hyperbolicity.
Advances understanding of the structure of dynamical systems with homoclinic classes.
Abstract
We prove that for generic diffeomorphisms, every expansive homoclinic class is hyperbolic.
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