We consider "conformal" parameterizations of random fractal spaces $X$ arising as limits of certain stochastic subdivision rules.

One motivation comes from the field of random geometry, where it is an important and difficult problem to understand this parameterization when $X$ arises from limits of random planar maps.

Deterministic versions of our model, and the analogous questions relating to conformal parameterizations, are closely related to W. Thurston's topological characterization of rational maps.

In all the settings mentioned above, a key difficulty is in proving that the fractal approximations do not degenerate in the complex analytic sense.

We will overcome this difficulty in our setting by proving a contraction inequality for probabilistic iteration on a variant of the universal Teichmuller space.

This inequality also provides a different perspective on random quasiconformal map models considered by Astala-Rohde-Saksman-Tao and Ivrii-Markovic.