Speaker: Natalia Jurga (University of St Andrews)
21 April, 2021 13:30 UTC
Abstract: A finite set of matrices $A \subset \mathrm{SL}(2,\mathbb{R})$ acts on one-dimensional real projective space $\mathbb{P}^1$ through its linear action on $\mathbb{R}^2$. In this talk we will be interested in the smallest closed subset of $\mathbb{P}^1$ which contains all attracting and neutral fixed points of matrices in $A$ and which is invariant under the projective action of $A$. Recently, Solomyak and Takahashi proved that if $A$ is uniformly hyperbolic and satisfies a Diophantine property, then the invariant set has Hausdorff dimension equal to the minimum of 1 and the critical exponent. In this talk we will discuss an extension of their result beyond the uniformly hyperbolic setting. This is based on joint work with Argyrios Christodoulou.