Jonathan Fraser, University of St. Andrews
"The Assouad spectrum"
The Assouad dimension is a familiar notion of dimension which, for a given metric space, returns the infimal exponent α ≥ 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r)^α balls of radius r. Motivated by this, we introduce a spectrum of dimensions designed to yield more information about the scaling structure of the space. More precisely, to each θ ∈ (0, 1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/ log r = θ. We discuss the resulting ‘dimension spectrum’ (as a function of θ) including its basic analytic and geometric properties as well as the precise calculation of the spectrum for some specific examples. This is joint work with Han Yu (St Andrews).
MC 5417