Intermediate dimensions of infinitely generated attractors

Abstract

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter θ ∈ [ 0 , 1 ] \theta \in [0,1] which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbański concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a ‘generic’ infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where ‘generic’ can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.

Attainable forms of intermediate dimensions

“…Let g 0 = lim y→0 + g(y) and note that g 0 ∈ (λ, α) by Lemma 3.2. Choose r 1 such that 2d log (2) log(1/r 1 ) = g 0 and let w 0 = log log(1/r 1 ). Now set x 1 = w 0 and, inductively, set x k+1 = ψ(x k ) for each k ∈ N. Let ρ k = exp(− exp(x k )) denote the corresponding scales, and set r k := ρ k /ρ k−1 for k ≥ 2.…”

“…For example, Bedford-McMullen carpets are a recent example of a natural family of sets for which the intermediate dimensions exhibit interesting properties [3]. Other sets which have been studied in the literature include infinitely generated self-conformal sets [2] and elliptical polynomial spirals [6]. However, in general, no progress has been made on determining sharpness of the general constraints on the intermediate dimensions.…”

Assouad type dimensions of infinitely generated self-conformal sets

“…Mauldin and Urbański [33, theorem 2.11] proved that the upper box dimension does indeed satisfy this formula. As the main result of [4], we proved that the upper intermediate dimensions dim θ also satisfy the naïve formula, that is, dim θ F = max{dim H F, dim θ {S i (x)}} for all θ ∈ [0, 1]. The intermediate dimensions are a family of dimensions (introduced in [15] and studied further in [3,6,7,9]) which are parameterised by θ ∈ [0, 1] and interpolate between the Hausdorff and box dimensions.…”

“…One of the most interesting features of the analysis in this paper is that the naïve formula does not always hold for the Assouad spectrum, which instead satisfies two bounds which can be sharp in general. In [4], we also proved bounds for the Hausdorff, box and intermediate dimensions of the limit set without assuming conformality or separation conditions. However, the Assouad dimension can be particularly sensitive to separation conditions even in the case of finite IFSs (see [18, section 7.2]), and in this paper we assume conformality and appropriate separation conditions throughout.…”

“…The dimension theory of infinite IFSs has been studied extensively, for example in [4,11,[31][32][33]42]. There are many similarities between finite and infinite iterated function systems, but also several differences.…”