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The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map?

A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ sufficiently $C¹$-close to $f$, then there exist a homoemorphism $h$ such that $f∘h=h∘g$. This is the topological conjugacy.

My question is: Why the map $g$ should be of class $C¹$. Why this definition does not works with for example $C⁰$ or a discontinuous map $g$ considered as a small variation of the map $f$.

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  • $\begingroup$ Because $C^0$ perturbations are too damaging. They can destroy the original structure of the system. $\endgroup$
    – Siming Tu
    Feb 28 '19 at 11:44
  • $\begingroup$ @SimingTu: Can you elaborate with this by giving some explanations with some references. $\endgroup$
    – Safwane
    Feb 28 '19 at 13:57
  • $\begingroup$ For example, consider a system with a hyperbolic fixed point, you can make a $C^0$ perturbation to make that in a small neighborhood of the fixed point, all points are fixed point. $\endgroup$
    – Siming Tu
    Mar 1 '19 at 0:33

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