Dimension theory of fractals on the circle
Vortrag von Yuxiang Jiao
Datum: 22.04.26 Zeit: 13.30 - 14.30 Raum: ETH HG G 19.1
By an elementary topological argument, any group of circle homeomorphisms that has no finite orbits and does not act minimally on the circle preserves a unique Cantor-like minimal set. The classical Patterson-Sullivan theory of Fuchsian groups fully describes the dimensional properties of these fractal sets whenever the group acts by \(\mathrm{PSL}_2(\mathbb R)\) elements. For more general smooth actions, a previous result by Deroin-Kleptsyn-Navas asserts that such a minimal set is always a Lebesgue null set in the real analytic category.
In this same setting, we establish a dimension theory for these fractal sets, which extends the classical Patterson-Sullivan theory to a broad smooth setting. In particular, we show that the Hausdorff and box dimensions of such minimal sets always coincide and are strictly less than one, which also improves upon the result of Deroin-Kleptsyn-Navas. This talk is based on joint work with Weikun He, Disheng Xu, and Wenyuan Yang.