Exceptional sets in dynamical systems and Diophantine approximation

Details Exceptional sets in dynamical systems and Diophantine approximation Exceptional sets in dynamical systems and Diophantine approximation

2001-08-30

arxiv.org/abs/math/0108210v1

Mathematics

Number Theory

Scientific paper

The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical properties of these exceptional sets are closely related to fundamental results in the metrical theory of Diophantine approximation. The counterpart of Diophantine approximation in hyperbolic space and a dynamical interpretation which led to the very general notion of `shrinking targets' are sketched and the recent use of flows in homogeneous spaces of lattices in the proof of the Baker-Sprindzuk conjecture is described briefly.

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