Rigidity and inflexibility in conformal dynamics (Q1126738)

The author presents a connection between the rigidity of hyperbolic 3-manifolds and universal scaling phenomena in dynamics. Starting with an inflexibility theorem for 3-manifolds of infinite volume and the notion of generalized Mostow rigidity, the author connects this inflexibility to dynamics. In Sections 2-6, the author discusses the geometrization of 3-manifolds which fiber over the circle; the extension of the inflexibility of Kleinian groups and their limit sets to certain other conformal dynamical systems and their Julia sets; the renormalization of unimodal maps; critical circle maps; and the self-similarity of Siegel disks. In Section 7, the author concludes with progress towards the classification of hyperbolic manifolds. The paper also contains a useful table which summarizes the parallels emerging between hyperbolic manifolds, quadratic-like maps on the interval, critical circle maps and Siegel disks.