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Journal of the London Mathematical Society 1994 49(3):614-624; doi:10.1112/jlms/49.3.614
© 1994 by London Mathematical Society
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© The London Mathematical Society

The Hausdorff Dimension of Exceptional Sets Associated with Normal Forms

M. M. Dodson, B. P. Rynne and J. A. G. Vickers

Department of Mathematics, University of York York YOl 5DD
Department of Mathematics, Heriot-Watt University Edinburgh EH1 1HX
Department of Mathematics, University of Southampton Southampton SO9 5NH

Received 30 October 1991. Revision received 7 July 1992.

The Hausdorff dimension is obtained for exceptional sets associated with linearising a complex analytic diffeomorphism near a fixed point, and for related exceptional sets associated with obtaining a normal form of an analytic vector field near a singular point. The exceptional sets 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 approximation and for which a lower bound for the Hausdorff dimension has been obtained.


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