Dimension rigidity in conformal structures

TL;DR

This paper establishes a dimension rigidity dichotomy for limit sets of conformal dynamical systems, showing they are either manifolds or fractals, with new techniques extending to infinite-dimensional cases.

Contribution

It introduces pseudorectifiability and extends dimension rigidity results to infinite-dimensional conformal systems, unifying various classes and weakening previous hypotheses.

Findings

Limit sets are either manifolds or fractals based on a dimension condition.

The new notion of pseudorectifiability handles infinite-dimensional cases.

Improves on previous rigidity results by weakening hypotheses.

Abstract

Let $\Lambda$ 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: $\Lambda$ 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 Urba\'nski ('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 \v{S}tan$'$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).