Past Events
One World Fractals, 2 November 2022
 DeJun Feng
(video).
 Dimensions of projected sets and measures on typical selfaffine sets.

Abstract: Let \(\{T_ix+a_i\}_{i=1}^\ell\) be an iterated function system on \({\Bbb
R}^d\) consisting
of invertible affine maps with \(\T_i\<1/2\),
and \(\pi:\{1,\ldots,\ell\}^{\Bbb N}\to {\Bbb R}^d\) the
corresponding coding map. For every Borel set \(E\) and
every Borel probability measure \(\mu\) in the coding space,
we determine the various dimensions of their projections
under \(\pi\) for typical translations \((a_1,\ldots,
a_\ell)\); in particular, we give a necessary and
sufficient condition on \(\mu\) so that the typical
projection of \(\mu\) is exact dimensional. This extends the
known results in the literature on typical projections of
invariant sets and invariant measures. It plays an
analogue to the classical theorems for fractal dimensions
under orthogonal projections.
The talk is based on joint work with ChiuHong Lo and CaiYun Ma.
 Aleksi Pyörälä
(video).
 On local structure of selfconformal measures.

Abstract:
A fundamental concept in analysis is that of tangent. Tangents capture the local
structure of an object but are often much more regular, and by studying the collection
of tangents at every point one can obtain information on global properties of the
object. A similar idea holds for measures: One can obtain very strong statements on
global geometric properties of a measure by studying the sequences of measures, called
"sceneries", obtained by "zooming in" around typical points for the measure. In our
recent work with Balázs Bárány, Antti Käenmäki and Meng
Wu, we study the sceneries of
selfconformal measures in the absence of any separation conditions. It turns out that
on the line, these measures are always uniformly scaling, meaning that the sceneries at
typical points share similar statistics almost everywhere. As one corollary, we obtain
that typical numbers for selfconformal measures are always normal in most bases.
 Amir Algom
(video).
 Fourier decay for selfsimilar measures.

Abstract: In recent years there has been an explosion of interest and progress on the
Fourier decay problem for fractal measures. We will survey some of these results and
remaining open problems, focusing on selfsimilar measures. We will also outline a
method to tackle this problem for smooth images of the measure, when the IFS in question
has noncyclic contraction ratios.
Joint work with Federico Rodriguez Hertz and Zhiren Wang.