Differentiability of \(p\)-central Cantor sets (Q1382779)

This paper provides, for each \(r\geq 1\), a geometrical construction of Cantor sets which are regular of class \(C^r\), and whose self-arithmetic difference is a Cantor set of positive Lebesgue measure. These Cantor sets are obtained by removing symmetrically \(p\) open intervals of the same length in each of the \((p+1)^n\) connected components of the \(n\)th order approximation \(I^n\) of the asymptotic limit. The lengths of the removed intervals change at each step and are ruled by a sequence \((r_i)_{i\in\mathbb{N}}\) of positive real numbers, such that the length of each connected component of \(I^n\) equals \(\prod^n_{i= 1}r_i\). Conditions on the sequence \((r_i)_{i\in\mathbb{N}}\) are given, providing regularity properties and the positiveness of the Lebesgue measure of the self-arithmetic difference. Furthermore, these Cantor sets are dynamically defined, when they are regular of class \(C^r\) with \(r\geq 2\). The introduction of fractal dimensions (Hausdorff dimension and limit capacity) shows that a small perturbation on the sequence \((r_i)_{i\in\mathbb{N}}\) yields Cantor sets whose self-arithmetic difference set either contains intervals or is a Cantor set of zero Lebesgue measure. The paper ends with the construction of a diffeomorphism on the sphere, with a basic set (a horseshoe) which is the product of one of these Cantor sets studied here, with itself.