The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

Abstract

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the 'upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the quasi-Assouad dimension at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu.We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.Mathematics Subject Classification 2010: primary: 28A80.

Lower Assouad Dimension of Measures and Regularity

“…Remark A.3. The analogous result was proved for the Assouad dimension in [4] for subsets of R d , but the same proof applies in any doubling metric space.…”

“…We also give an example to show that equality of the upper and lower Assouad dimensions does not imply s-regularity of the measure. In analogy with what was shown for sets in [2] and [4], we prove that the quasi-lower and quasi-upper Assouad dimensions of measures can be recovered from the Assouad dimension spectrum of a measure under the assumption that the measure is quasidoubling, i.e., has finite quasi-upper Assouad dimension. These results can all be found in Sections 2 and 6.…”

“…A µ for all n ∈ N. These observations, together with the continuity result of the previous lemma, allow one to use the same argument as given directly after Lemma 3.1 in [4] to show that lim inf θ→1 dim =θ A µ = lim θ→1 dim =θ A µ and similarly for the upper Assouad spectrum. That completes the proof of Theorem 6.2.…”

“…In [4] it was shown that h(1/θ − 1) = sup 0<ψ≤θ dim =ψ A E for subsets of R d , although the same proof holds for any doubling metric space E, i.e. spaces where dim A E < ∞.…”

“…However, it is possible to obtain the same conclusion for measures which only satisfy the weaker (quasi-doubling) condition, dim qA µ < ∞ and this we do in Theorem 6.2 below. The general scheme of the proof is essentially the same as in [4], but new technical complications arise. Examples of such measures include equicontractive, self-similar measures that are regular, meaning the probabilities associated with the right and left-most similarities are equal and minimal.…”