Tuomas Sahlsten: Homology and chaotic quasiregular dynamics

Speaker: Tuomas Sahlsten

Title: Homology and chaotic quasiregular dynamics

Abstract: Quasiregular maps are a good candidate for generalisations of holomorphic maps in general Riemannian manifolds. They have deep connections to the theory of differential forms and p-harmonic functions on the manifolds, and form an important part in the theory of distributional limits of random structures (graphs, manifolds, etc.).

We are interested to do dynamics for quasiregular maps self-maps of a manifold M. However, unlike holomorphic maps the distortions of QR maps may not stay under control when one iterates so here it helps to assume the QR maps are Uniformly Quasiregular (UQR). These maps play the role of rational maps in holomorphic dynamics. It turns out that having “non-trivial” UQR dynamics on the manifold M greatly influences the shape (topology) of the manifold M. For example a recent result by Kangasniemi shows that as long as the manifold M is not a rational homology sphere, the Julia set of any non-trivial UQR map has positive volume! This is a bit shocking since in holomorphic dynamics Julia sets usually have fractal dimension. In this talk we use the methods of Kangasniemi and the theory of normal currents from Geometric Measure Theory (Federer et al.) to establish Shub’s entropy conjecture for UQR maps on non-rational homology spheres. This talk is based on a joint work with Ilmari Kangasniemi (Helsinki), Yusuke Okuyama (Kyoto) and Pekka Pankka (Helsinki).