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 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 ∈ (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.

Intermediate dimensions

“…A related approach to 'dimension interpolation' was recently considered in [5] where a new dimension function was introduced to interpolate between the box dimension and the Assouad dimension. In this case the dimension function was called the Assouad spectrum, denoted by dim θ A F (θ ∈ (0, 1)).…”

Lower Assouad-Type Dimensions of Uniformly Perfect Sets in Doubling Metric Spaces

“…Fractal set is a major research object in the nonlinear science. The lower Assouad dimension, introduced by Larman [14,15], is a tool to describe the local scaling properties of a set, which is a natural dual to the well-studied Assouad dimension, see [5,8,9,16] etc. They have played an important role in the Lipschitz embedding problems of metric spaces, dimension theory and homogeneity of fractals, details can found in [5,7,16] and references therein.…”

“…It can be seen from the definition that the lower Assouad dimension depends on two independent scales r and R, but it tells no information on which scales witness the minimal exponential growth rate. To treat this problem and see how the gauge depends on the scales, Fraser and Yu [8] introduced Thus, for each fixed θ ∈ (0, 1), we get a 'restricted' version of the lower Assouad dimension by letting the scales satisfy the relationship r = R 1 θ . Then one can vary θ and obtain a spectrum of dimensions which gives finer scaling information on the local structure of a set.…”

“…Besides, the lower Assouad spectrum is bi-Lipschitz and quasi-Lipschitz invariant for any θ ∈ (0, 1). One can refer to [5,8,9] for more details on properties of lower Assouad dimension and lower Assouad spectrum.…”

Assouad Spectrum Thresholds for Some Random Constructions

“…In their landmark paper [FY18], Fraser and Yu discuss the possibility of extending the definition of the Assouad spectrum to analyse the case when quasi-Assouad and Assouad dimension differ. These general spectra, which we shall also refer to as generalised Assouad spectra, would then shed some line on the behaviour of 'in-between' scales.…”