About $C^{1}$-minimality of the hyperbolic Cantor sets

TL;DR

This paper proves that certain smooth hyperbolic Cantor sets on the circle, near affine sets, cannot be minimal under $C^{1}$ diffeomorphisms, highlighting limitations in their dynamical complexity.

Contribution

It establishes that $C^{1+eta}$ hyperbolic Cantor sets close to affine sets are not $C^{1}$-minimal, revealing new constraints on their minimality properties.

Findings

Hyperbolic Cantor sets near affine sets are not $C^{1}$-minimal.

The result applies to $C^{1+eta}$ regularity, not just $C^{1}$.

Provides insight into the structure of minimal sets in smooth dynamics.

Abstract

In this work we prove that a $C^{1+\alpha}$-hyperbolic Cantor set contained in $S^1$, close to an affine Cantor set, is not $C^{1}$-minimal.