New dimension spectra: finer information on scaling and homogeneity

Abstract

We introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent $α\geq 0$ 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 $r$. To each $θ\in (0,1)$, we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales $r$ and $R$ used in the definition satisfy $\log R/\log r = θ$. The resulting `dimension spectrum' (as a function of $θ$) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding `lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension and the provision of new bi-Lipschitz invariants.