02/13/2025
By Joris Roos
Title: Continued fractions and the geometry of conformally invariant sets
Time: 11 a.m. to Noon
Room: Southwick Hall, Room 350W
Everyone is welcome!
Abstract: A well-known family of sets with non-integer dimension is obtained as follows: consider points in the interval [0,1] whose base-b expansion only contains digits in some (non-empty, proper) subset of {0, ... , b-1}. The prototypical example of such a set is the well-known middle-thirds Cantor set. While such sets are pathological in a classical sense, in the grand scheme of fractal sets, they are very nice sets. As we will see, one explanation for their nice properties is that these 'missing digit' sets are invariant for a well-behaved dynamical system (integer multiplication modulo 1) on a compact manifold (the torus). But what happens if we remove compactness? In this case, the canonical example is analogous to the above construction, except with the continued fraction expansion in place of the base-b expansion. The dynamics are still very well-behaved, but we lose compactness.
In this talk I will give a gentle introduction to the theory of conformally invariant sets; and to discuss the myriad of ways in which things go wrong without compactness. Most results are old; any new results are based on joint work with Amlan Banaji (University of Loughborough).
More information about the Department of Mathematics and Statistics colloquium.