Generalised intermediate dimensions

Abstract

We introduce a family of dimensions, which we call the Φ-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. We do this by restricting the allowable covers in the definition of Hausdorff dimension, but in a wider variety of ways than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We investigate relationships between the Φ-intermediate dimensions and other notions of dimension, and study many analytic and geometric properties of the dimensions. We prove continuity-like results which improve similar results for the intermediate dimensions and give a sharp general lower bound for the intermediate dimensions that is positive for all θ ∈ (0, 1] for sets with positive box dimension. We prove Hölder distortion estimates which imply bi-Lipschitz stability for the Φ-intermediate dimensions. We prove a mass distribution principle and Frostman type lemma, and use these to study dimensions of product sets, and to show that the lower versions of the dimensions, unlike the the upper versions, are not finitely stable. We show that for any compact subset of an appropriate space, these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of sets for which the intermediate dimensions are discontinuous at θ = 0, thus providing more refined geometric information about such sets.

Intermediate dimensions of infinitely generated attractors

“…Therefore letting (2) By Lemma 3.1 we may assume that Φ is invertible. Then (2) holds by the same proof as (1), with dim θ , δ θ and (δ/R w ) θ replaced by dim Φ , Φ −1 (δ) and Φ −1 (δ/R w ) respectively. In place of (3.4), R w Φ −1 (δ/R w ) Φ −1 (δ) holds since Φ(δ)/δ 0 monotonically as δ → 0 + by assumption.…”

“…(3) The proof is motivated by the proof of [29, Lemma 2.8], which gives a result for the box dimension in the less general setting of a CIFS. We will consider δ ∈ 1 n+1 , 1 n and induct on n. The idea is that if we fix a large enough q ∈ N, the level-q cylinders with size δ can be covered efficiently using a cover of P q , and the cylinders with size δ can be covered efficiently using images of efficient covers of F with larger diameters that are assumed to exist by the inductive hypothesis, and the fact that P (s) < 0 if s > h.…”

“…The intermediate dimensions of the limit set of a CIFS will not generally be continuous at θ = 0. For example, if the fixed points are arranged to be at 1 log n then by Falconer, Fraser and Kempton's [12, Example 1], the intermediate dimensions of the fixed points will be 1 for all θ ∈ (0, 1], but the contraction ratios can be made small enough (and tending to 0 rapidly enough) so that h < 1. In this case, the limit set and its closure will have the same Hausdorff dimension as they differ by a countable set (namely the images of the point 0 under maps corresponding to finite words) by Mauldin and Urbański's [30,Lemma 2.1].…”

“…In this paper we study the intermediate dimensions of limit sets of infinite iterated function systems. The intermediate dimensions have been studied further in [2,3,4,10,15,27,34] and have been generalised to the Φ-intermediate dimensions by Banaji [1] to give more refined geometric information about sets for which the intermediate dimensions are discontinuous at θ = 0, which by Theorem 3.5 can happen for the limit sets studied in this paper (see the discussion after Theorem 4.3). The intermediate dimensions are an example of a broader notion of 'dimension interpolation' (see the survey [15]), which seeks to find a geometrically natural family of dimensions which lie between two familiar notions of dimension.…”

“…The cube [0, 1] d is clearly a weak tangent to (the closure of) G p,d and so the Assouad dimension of G p,d is d (see [14,Theorem 5.1.2]). By Banaji's general lower bound [1,Proposition 3.10],…”