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 $$f$$.
• Because $C^0$ perturbations are too damaging. They can destroy the original structure of the system. Feb 28 '19 at 11:44
• 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. Mar 1 '19 at 0:33