2015/04/30 by Tushar Das, David Simmons, Mariusz Urbański · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Topology and Set Theory #Caveolin-1 and cellular processes #Conformal map #Fractal #Hausdorff dimension #Iterated function #Iterated function system #Julia set #Kleinian group #Limit (mathematics) #Limit set #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Rigidity (electromagnetism) #math.DS #math.GT
paper · pdf · doi:10.1016/j.aim.2016.12.034
published as Adv. Math. 308 (2017), 1127-1186
arxiv created 2016/12/29 · openalex publication_date 2017/01/25 · arxiv updated 2018/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Let Λ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than the radial limit set, for the following dichotomy: Λ is either a real-analytic manifold or a fractal in the sense of Mandelbrot (i.e. its Hausdorff dimension is strictly greater than its topological dimension). Our primary focus is on the infinite-dimensional case. An important component of the strategy of our proof comes from the rectifiability techniques of Mayer and Urbański ('03), who obtained a dimension rigidity result for conformal iterated function systems (including those with infinite alphabets). In order to handle the infinite dimensional case, both for Kleinian groups and for iterated function systems, we introduce the notion of pseudorectifiability, a variant of rectifiability, and develop a theory around this notion similar to the theory of rectifiable sets. Our approach also extends existing results in the finite-dimensional case, where it unifies the realms of Kleinian groups, conformal iterated function systems, and rational functions. For Kleinian groups, we improve on the rigidity result of Kapovich ('09) by substantially weakening its hypothesis of geometrical finiteness. Moreover, our proof, based on rectifiability, is entirely different than that of Kapovich, which depends on homological algebra. Another advantage of our approach is that it allows us to use the "demension" of Štan'ko ('69) as a substitute for topological dimension. For example, we prove that any dynamically defined version of Antoine's necklace must have Hausdorff dimension strictly greater than 1 (i.e. the demension of Antoine's necklace).