Past Events
One World Fractals, 22 March 2023

Vilma Orgoványi
(video).
 Overlapping random selfsimilar sets on the line.

Abstract: The Mandelbrot percolation fractal in \(\mathbb{R}^d\) is constructed
inductively as follows. We fix an integer \(M\geq 2\) and a probability
\(p\in(0,1)\).
The closed unit cube is divided into \(M^d\) congruent subcubes, each of which is
independently retained with probability \(p\) and discarded with probability \(1p\).
This process is repeated in the retained cubes ad infinitum, or until there
are no cubes left. The Mandelbrot percolation process restricted to the building
blocks of a sponge yields a random sponge. We consider the rational projections of
random sponges into lines from the perspective of Hausdorff dimension, positivity of
Lebesgue measure and existence of interior points. In contrast to Mandelbrot
percolation, the inhomogeneity of the sponge yields qualitatively different
phenomena.
This is based on joint work with Károly Simon.

Pablo Shmerkin (video).
 Improved bounds for the dimensions of Furstenberg sets.

Abstract:
An important and multifaceted problem in geometric measure theory is understanding
incidences between \(\delta\)tubes and \(\delta\)balls. The Furstenberg set problem is
a concrete instance of this problem. I will present some \(\epsilon\)improvements on
the previously known bounds for Furstenberg sets, obtained in collaborations with T.
Orponen and with H. Wang.

Alec Chamberlain Cann (video).
 More refined multifractal spectra for Bedford McMullen carpets.

Abstract: We study selfaffine measures on Bedford McMullen carpets. The
multifractal spectrum assesses the size of sets of points for which the local
dimension takes a prescribed value. We make a refinement to the known spectrum that
takes into account points for which the local dimension does not necessarily exist
by considering upper and lower symbolic local dimension. For carpets on a \(n \times
2\)
grid, and the measure that assigns mass uniformly on rectangles, we exhibit
different behaviour from that seen when the refined spectrum is computed for
selfsimilar sets.
 Organisers: Thomas Jordan and Henna Koivusalo
One World Fractals, 18 January 2023

Amlan Banaji
(video).
 Dimensions of selfaffine carpets.

Abstract: For selfaffine carpets, different notions of fractal dimension such as
Hausdorff, box and Assouad dimension often take different values. In the setting of
BedfordMcMullen carpets, we will describe some of the methods used to calculate
these dimensions, as well as two other dimension spectra, namely the Assouad
spectrum and intermediate dimensions, which provide more refined local and global
geometric information about the sets. For more general carpets of LalleyGatzouras
type, the Assouad spectrum can display more interesting behaviour. This talk is
based on joint work with Jonathan Fraser and István Kolossváry.

Hong Wang (video).
 Restricted projections in R^d.

Abstract: We prove a Marstrandtype projection theorem for directions in a
nondegenerate curve. The proof uses BourgainDemeter and BourgainDemeterGuth
decoupling theorems. This is based on joint work with Gan and Guo.
 Organisers: Jonathan Fraser and Natalia Jurga
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.
 Organisers: Antti Käenmäki and Sascha Troscheit