John Gill (climber) - Mathematical Research - Highlights

Highlights

Gill began studying the convergence behavior of infinite compositions of linear fractional transformations in the late 1960s. After establishing sufficient conditions for convergence in the parabolic and elliptic cases he discovered a way to accelerate convergence of limit periodic continued fractions using an attractive fixed point, then a way to analytically continue certain continued fractions using a repulsive fixed point. He developed initial theory on the convergence of more general infinite compositions satisfying contractive conditions, leading to a paper by L. Lorentzen describing convergence of sequences of forward (or right or inner) compositions of analytic functions that uniformly contract into a compact subset of a simply connected domain. Gill then developed a similar theory for left (or backward or outer) compositions of analytic functions under similar hypotheses. Lorentzen’s result may be applied, for example, to the analytic theory of continued fractions and Gill’s result to the evaluation of fixed points of functions defined by infinite expansions, or convergence of reverse continued fractions. Both of these theorems can be considered extensions of Brouwer's fixed point theorem for analytic functions.