Cantor sets constructed by infinitely renormalizable diffeomorphisms
The author investigates the notion of infinitely renormalizable diffeomorphisms in the context of two-dimensional dynamics. Constructions of such diffeomorphisms have been previously discussed by Bowen, Franks and Young. This paper establishes that the regularity of these diffeomorphisms is closely related to the Hausdorff dimension of their attracting, invariant Cantor sets. Indeed, if \(K\) is such a Cantor set and \(f: D\to D\) is an infinitely renormalizable diffeomorphism of the unit disk in \(\mathbb{R}^2\) with \(f\in C^{1+\varepsilon}\), then \(\varepsilon\leq \dim_H(K)\) where \(\dim_H\) denotes the Hausdorff dimension. Moreover, the author shows that if \(f\in C^{2+\varepsilon}\), then \(1+\varepsilon\leq \dim_H(K)\). Furthermore, there are no infinitely renormalizable diffeomorphisms of class \(C^3\).
infinitely renormalizable diffeomorphisms
scientific article; zbMATH DE number 943250