### Julia sets with Ahlfors-regular conformal dimension one by InSung Park

- Series
- Geometry Topology Seminar
- Time
- Monday, February 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE
- Speaker
- InSung Park – Indiana University Bloomington – park433@iu.edu

**Please Note:** Office hours will be held 3-4pm EST.

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map *f*, the Julia set *Jf* is a beautiful fractal defined as the repeller of the dynamics of *f*. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an *f*-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.