Strongly automorphic mappings

Alastair Fletcher (Northern Illinois University)

A result of Ritt highlights the connection between periodic holomorphic functions and certain rational maps: powers of z, Chebyshev polynomials and Lattes rational maps. This connection arises through solutions to Poincare and Schroeder equations. We will introduce quasiregular mappings as higher dimensional analogues of holomorphic functions and then discuss the quasiregular counterparts of the Ritt result. This will include: (i) constructions of strongly automorphic mappings, (ii) a classification in terms of crystallographic orbifolds for which automorphism groups give rise to such mappings, (iii) showing that solving the Schroeder functional equation yields a uniformly quasiregular map whose Julia set is either a quasi-sphere, quasi-disk or all of space, (iv) time allowing, we will discuss to what extent the converse can apply and a three dimensional Denjoy-Wolff Theorem. This talk is based on joint work with Doug Macclure (NIU).