Let μ be an invariant measure supported on the repeller of a uniformly expand-
ing planar map T . While the local scaling properties of the measure μ at typical
points are closely related to its dimension, one is often interested in understanding
the scaling of the measure at non-typical points. The multifractal formalism is a
type of duality relating the sets of points with given scaling properties with certain
statistical smoothness estimates (the Lq-spectrum and related constructions) for
the measure μ.
If T is conformal, then the underlying optimization problem satisfies a weak
form of convexity and the situation is well-understood. However, without con-
formality, much less is known. In fact, through some explicit examples, we will
see that a wide range of exotic behaviour is possible such strong non-convexity
properties, non-differentiability of the Lq-spectrum, and breakdown of duality
and the failure of the multifractal formalism. This is based on joint work with
Thomas Jordan (Bristol) and István Kolossváry (St Andrews).