| 13:00 - 13:30 |
Karoly Simon: (Budapest University of Technology and Economics) Randomly perturbed self-similar sets ↓ We are given a self-similar Iterated Function System (IFS) on the real line \mathcal{S}:=\{S_1,\dots, S_m\}.
We fix a sufficiently large interval \widehat{I} which is sent into itself by all mappings of \mathcal{S}.
For an arbitrary n \geq 1, and \mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n the corresponding level n cylinder interval is
\begin{equation}
\label{2}
I_{ i_1,\dots,i_n }: =S_{i_1}\circ\dots\circ S_{i_n}(\widehat{I}).
\end{equation}
The collection of level n
cylinder intervals is
\mathcal{I}_n
:=
\left\{
I_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n
\right\}.
The attractor is
\begin{equation}
\label{1}
\Lambda :=\bigcap _{n=1}^{\infty }\bigcup_{I\in\mathcal{I}_n}I.
\end{equation}
We say that \Lambda
is self-similar set.
\bigskip
{\bf{Open Problem}}
Is there a self-similar set of positive Lebesgue measure and empty interior on the line?
\bigskip
We consider this problem for randomly perturbed self-similar sets, which
are obtained in the following way:
In the randomly perturbed case, the
n-cylinder interval \widetilde{I}_{ i_1,\dots,i_n } corresponding
to the indices \mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n is
obtained
by replacing S_{i_k} in formula \eqref{2}
by a random and independent of everything translation \widetilde{S}_{i_k}
of S_{i_k} for all k=1,\dots, n. Then we build the randomly perturbed attractor
in an analogous way to formula \eqref{1} from the randomly perturbed cylinder intervals \widetilde{I}_{i_1\dots i_n}. That is
\widetilde{\mathcal{I}}_n
:=
\left\{
\widetilde{I}_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n
\right\},
and the randomly perturbed self-similar set is
\widetilde{\Lambda }:=
\bigcap _{n=1}^{\infty }\bigcup_{\widetilde{I}\in\widetilde{\mathcal{I}}_n}\widetilde{I}.
First, I review results related to the Lebesgue measure and
Hausdorff dimension of these randomly perturbed self-similar sets.
Then, I turn to our new result (joint with M. Dekking, B. Szekely, and N. Szekeres)
about the existence of interior points in these randomly perturbed self-similar sets. (TCPL 201) |
| 13:48 - 13:53 |
Caleb Marshall: (UBC) A Projection Theoretic Triptych ↓ We spend our small time sharing with the audience three questions in projection theory. Each of these correspond to the ongoing research program of the presenter. The idea is to give a brief sketch (largely in pictures and examples) of the main questions, with a few comments on the tools utilized to examine each problem. Short problem descriptions are given below.
The Favard Length problem (1) asks: if a set in the plane has positive and finite length, but almost-every orthogonal projection of the set has zero length, what is the expected asymptotics of projections of thin neighbourhoods of the set? Continuous Erdős-Beck Theorems (2) assert that, whenever sets are not concentrated (in a dimensional sense) on hyperplanes, then the associated family of spanning lines must have complementarily-large dimension. Falconer-Type Estimates for Dot Products (3) asks one to produce a dimensional threshold for sets in n-dimensional Euclidean space such that the associated dot product set has positive Lebesgue measure.
Material for Problem (1) is based on past/ongoing joint work with Izabella Łaba; Material for Problem (2) is based on ongoing joint work with Paige Bright; Material for Problem (3) is based on ongoing joint work with Paige Bright and Steven Senger. (TCPL 201) |
| 13:54 - 13:59 |
Alex McDonald: (OSU) Point configurations in products of thick Cantor sets ↓ An area of much recent activity in geometric measure theory is the study of conditions on compact sets which guarantee the existence of patterns. A classic example is the Falconer distance problem, which asks how large the Hausdorff dimension must be to ensure a positive measure worth of distances (or, more generally, to ensure the set of distances contains a non-degenerate interval). In this talk I will discuss analogous problems where the notion of Hausdorff dimension is replaced with "thickness", another natural notion of size for Cantor sets. (TCPL 201) |
