## Past Events

### One World Fractals, 2 November 2022

• De-Jun Feng (video).
• Dimensions of projected sets and measures on typical self-affine 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 Chiu-Hong Lo and Cai-Yun Ma.

• Aleksi Pyörälä (video).
• On local structure of self-conformal 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 self-conformal 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 self-conformal measures are always normal in most bases.

• Amir Algom (video).
• Fourier decay for self-similar 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 self-similar measures. We will also outline a method to tackle this problem for smooth images of the measure, when the IFS in question has non-cyclic contraction ratios.
Joint work with Federico Rodriguez Hertz and Zhiren Wang.