Dimensions of fractional Brownian images

Abstract

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Capacity 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-α fractional Brownian motion in terms of capacity theoretic dimension profiles. As a corollary, this establishes continuity of the profiles for all Borel sets, allowing 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 in the setting of projections.

Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

“…Intuitively, a dimension gap occurs when the set is inhomogeneous in space, and thus cannot be covered optimally by using sets of equal diameter. There are many sets in R d which exhibit this phenomena: for example convergent sequences, elliptical polynomial spirals, attenuated topologists' sine curves; or dynamically defined attractors such as limit sets of infinite conformal iterated function systems, self-affine carpet-like constructions in the plane and sponges in higher dimensions; or images of sets under certain stochastic processes [9,13]; or even the connected component of supercritical fractal percolation [8].…”

“…Intermediate dimensions can also be formulated using capacity theoretic methods [10]. This approach has been used to bound the Hölder regularity of maps that can deform one spiral into another [11], compute the almost-sure value of the intermediate dimension of the image of Borel sets under index-α fractional Brownian motion in terms of dimension profiles [9] and under more general Rosenblatt processes [13].…”

“…and dim θ f (F ) ≤ α −1 dim θ F (see [9,Theorem 3.1] and [1,Section 4] for further Hölder distortion estimates). In [2, Example 4.5], Banaji and Fraser showed that the intermediate dimensions can…”

“…give better information about the Hölder distortion between some continued fraction sets than the Hausdorff or box dimensions. The intermediate dimensions can also have applications to the orthogonal projections of a set, see [10,Section 6], and images of a set under index-α fractional Brownian motion, see [9,Section 3]. This relies on the intermediate dimensions of the set in question being continuous at θ = 0.…”

Intermediate dimensions of infinitely generated attractors

“…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.…”