Alex Rutar, Department of Pure Mathematics, University of Waterloo
"Geometric and Combinatorial Separation Conditions for Iterated Function Systems"
In our context, an iterated function system (IFS) is a finite set of maps from $\mathbb{R}$ to $\mathbb{R}$ of the form $\{x\mapsto r_i x+d_i\}_{i=1}^m$ with $|r_i|<1$. Any IFS has an associated invariant compact set $K$ (for example, the Cantor set) and a family of invariant measures supported on $K$. It is a classical question in fractal geometry to understand dimensional properties of these invariant objects; however, this is challenging to do in full generality. As a result, we will discuss two well-studied separation conditions for IFSs: one with a geometric flavour, and another with a combinatorial flavour. This talk will include recent work on the relationship between these separation conditions, as well as the dimension theory of IFSs which satisfy them. This is joint work with Kathryn Hare and Kevin Hare.
MC 5417