Projective Spectrum, Self-similarity, and Complex Dynamics

Rongwei Yang

Finitely generated structures are important subjects of study in various mathematical disciplines. Examples include finitely generated groups, Lie algebras and C^∗-algebras, etc. It is thus a fundamental question whether there exists a universal mechanism in the study of these vastly different entities. In 2009, the notion of projective spectrum for several elements A_1, ..., A_n in a unital Banach algebra B was defined through the multiparameter pencil A(z) = z_1 A_1 + · · · + z_n A_n, z ∈ C^n. This conspicuously simple definition turned out to have a surprisingly rich content. This talk briefly reviews some recent results on projective spectrum and then focuses on its connection with self-similar group representations and complex dynamics. In particular, we will see how the projective spectrum and Julia set are coupled through the self-similarity of the infinite dihedral group. Some details of the proof will be given, and an application of the result to the Grigorchuk group will be mentioned as well.

http://scgp.stonybrook.edu/video_portal/video.php?id=4427

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory