Exceptional Sets in Dynamical Systems and Diophantine Approximation

Abstract

Abstract. 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-Sprindžuk conjecture is described briefly.

Renewal of singularity sets of random self-similar measures

“…This achievement is a non-trivial generalization of what is referred to as "ubiquity properties" (see [19] and references therein) of the resonant system ([2]) {(x n , λ n )} n .…”

“…In this homogeneous context, the measure µ is the Lebesgue measure, and computing the dimension of these sets relies on the notion of ubiquity (see [19] for instance). The study of X • µ([0, t]) 0≤t≤1 necessitates the notion of heterogeneous (or conditioned) ubiquity introduced in [12,10].…”

Renewal of singularity sets of random self-similar measures

“…The fractal geometry of supremum limit sets already occupies an important role in determining the multifractal nature of homogeneous sums of Dirac masses [22] and Lévy processes [28]. In this homogeneous context, the measure µ is the Lebesgue measure, and computing the dimension of these sets relies on the notion of ubiquity (see [19], for instance). The study of (X • µ([0, t])) 0≤t≤1 requires the notion of heterogeneous (or conditioned) ubiquity introduced in [10] and [8].…”

The Geometry of Farey Staircases

“…These classifications will be compared and studied below. In [Dodson, 2001] we have a survey of results extending these concepts to higher dimensions and to non-Euclidean spaces.…”