OU Dynamical Systems seminar

The Dynamical Systems group at the Open University is organising an online seminar series.  

Our group's interests include -but are not restricted to:  Aperiodic Order,  Computer-assisted proofs in dynamical systems,  Energy systems modelling,  Holomorphic Dynamics,  Hyperbolic Geometry and Cocycle Dynamics,  Renormalisation in dynamical systems and its applications,  Symbolic Dynamics,  Tiling Dynamical Systems,  and Topological Dynamics.

Upcoming seminars

• Wednesday 30 September 2.00pm

Speaker: Petra Staynova, University of Derby

Title: Another Naturally Appearing Family of Cantorvals

Abstract:  A Cantorval is an intriguing topological subset of the real line that combines the properties of both an interval and a Cantor set. While Cantorvals have been observed in the study of sumsets of Cantor sets, their systematic construction through digit systems remains relatively unexplored.

In this paper, we investigate a family of self-similar sets generated by standard primitive digit systems (Q, D). We focus specifically on the family Q=3 with digit sets D = {-1, 0, 3k+1} for k a natural number. We classify the topology of these digit tiles, and prove that if the digit set D consists of consecutive integers, the resulting tile is an interval. Otherwise, if Q is prime and D consists of non-consecutive integers, the resulting tile is a symmetric Cantorval.

Using the contact matrix method of Grochenig and Haas, we provide a closed-form representation for the matrices describing the boundary of these tiles under some mild assumptions. We calculate the Hausdorff dimension of the boundary, $\dim_H \partial t_k$, and provide numerical evidence about the behavior of the dimension as $k$ approaches infinity. This work provides a new, constructive framework for generating and analyzing Cantorvals in the context of fractal tilings.

This is joint work with Natalie Frank, Anton Gorodetski, Jan Mazac, May Mei, and Kitty Yang.

Past seminars and seminar schedule