Chairperson, Graduate Committee: Lukas GeyerManlove, Joseph Michael2016-01-032016-01-032015https://scholarworks.montana.edu/handle/1/9061The results presented here answers in part a conjecture of Douady about sharpness of the Brjuno condition. Douady hypothesized that a Siegel disk exists for a rational function if and only if the Brjuno condition is satisfied by the rotation number. It is known that the Brjuno condition is sharp for quadratic polynomials and many special families. This thesis focuses on a class of rational functions, many of which have not been considered previously. Specific examples of maps for which these results apply include quadratic rational maps with an attracting cycle. Also included are those rational functions arising of Newton's method on cubic polynomials with distinct roots.enFatou setsRational equivalence (Algabraic geometry)HomeomorphismsDissertationCopyright 2015 by Joseph Michael Manlove