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, meaning that the Hausdorff dimension of $\Lambda$ 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 {\bf pseudorectifiability}, a
variant of rectifiability, and develop a theory around this notion
similar to the theory of rectifiable sets.

These results are joint-work with David Simmons (Ohio State) and
Mariusz Urba\'nski (North Texas).