THE HAUSDORFF DIMENSION OF EXCEPTIONAL SETS ASSOCIATED WITH NORMAL FORMS

The Hausdorff dimension is obtained for exceptional sets associated wi th linearising a complex analytic diffeomorphism near a fixed point, a nd for related exceptional sets associated with obtaining a normal for m of an analytic vector field near a singular point. The exceptional s ets consist of eigenvalues which do not satisfy a certain Diophantine condition and are 'close' to resonance. They are related to 'lim-sup' sets of a general type arising in the theory of metric Diophantine app roximation and for which a lower bound for the Hausdorff dimension has been obtained.