Dimensions of Fractional Brownian Images

Abstract

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

The fractal structure of elliptical polynomial spirals

“…Intuitively, the m-dimensional profile may be thought of as the dimension of an object when viewed from an m-dimensional viewpoint. In favour of brevity we omit a thorough introduction to dimension profiles, which may be found in [1,2,5,6], since the sole property we require is their relationship to fractional Brownian images [1,5,22]. In the following lemma, we bound the 2α-profiles of S p,q , denoted dim 2α θ S p,q , by a quantity strictly less than the dimension for all θ > 0, see Figure 6.…”

“…Intermediate dimensions have already seen surprising applications and properties, despite their recent introduction. For example, they have been used to establish relationships between the Hausdorff dimension of a set and the typical box dimension of fractional Brownian images [1] or orthogonal projections [2]. Other notable works include [16].…”

“…Figure 2. A plot of dim θ S p,q (y-axis) against θ (x-axis) for θ ∈ [0, 1] and dim θ−1 A S p,q against θ for θ ∈ [1,2]. In this example, p = 0.1 and q = 0.8.…”

“…Our methodology is based on the dimension profiles of Falconer and Howroyd [8] and recent works [1,5,6]. Of course, if there is an α-Hölder map between S p and S q we immediately obtain…”

“…The definition of the profiles is potential-theoretic and rather challenging to compute in the case of S p,q . This difficulty is circumvented by instead using their relationship [1,Theorem 3.4] to fractional Brownian images. In fact, the method employed here may be used more generally to estimate the Hölder regularity of a function between any two sets for which the box or intermediate dimensions of the fractional Brownian images may be estimated from above.…”