We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic
form of" self-similarity" under the operation of re-scaling, the dimension of linear images of …

We prove the following conjecture of Furstenberg (1969): if A,B⊂0,1 are closed and
invariant under *p\mathrmmod\1 and \times_q\mathrmmod\1, respectively, and if …

We give a fractal-geometric condition for a measure on 0, 1 0, 1 to be supported on points xx
that are normal in base nn, ie such that {n^ kx\} _ k ∈ N nkx k∈ N equidistributes modulo 1 …

Let $\nu $ be a self similar measure on $\mathbb {R}^ d $, $ d\geq 2$, and let $\pi $ be an
orthogonal projection onto a $ k $-dimensional subspace. We formulate a criterion on the …

We show that any self-conformal measure $\mu $ on $\mathbb {R} $ is uniformly scaling and
generates an ergodic fractal distribution. This generalizes existing results by removing the …

We study fractal measures on Euclidean space through the dynamics of" zooming in" on
typical points. The resulting family of measures (the" scenery"), can be interpreted as an orbit …

Let µ be a probability measure on [0, 1] which is invariant and ergodic for Ta (x)= ax mod 1,
and 0< dim µ< 1. Let f be a local diffeomorphism on some open set. We show that if E⊆ R …

We analyze the local geometric structure of self-similar sets with open set condition through
the study of the properties of a distinguished family of spherical neighborhoods, the typical …

We study the scaling scenery and limit geometry of invariant measures for the non-conformal
toral endomorphism (x, y)↦(mx mod 1, ny mod 1) that are Bernoulli measures for the natural …

This dissertation concerns the geometry of self-similar, self-affine and self-conformal sets
and measures in Euclidean spaces. The aim is to determine whether certain geometric and …