Tuomas Sahlsten

Dimension, entropy and the local distribution of measures

We present a general approach to the study of the local distribution of mea- sures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local homogeneity, porosity and conical densities of measures.

Dynamics of the scenery flow and geometry of measures

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

Structure of distributions generated by the scenery flow

We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery ow. We further show that the collection of fractal distributions is closed under the weak topology and, moreover, is a Poulsen simplex, that is, extremal points are dense. We apply these to show that a Baire generic measure is as far as possible from being uniformly scaling: at almost all points, it has all fractal distributions as tangent distributions.

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdiere. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

Tangent measures of non-doubling measures

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.

RADIAL PROJECTIONS OF RECTIFIABLE SETS

We show that if no

STABILITY AND PERTURBATIONS OF COUNTABLE MARKOV MAPS

m

Fourier transforms of Gibbs measures for the Gauss map

-plane contains almost all of an

Scaling scenery of (×m,×n) invariant measures

m

Scaling scenery of {

-rectifiable set