Fractal dimensions of the Rosenblatt process

Abstract

The paper concerns the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. We obtain results regarding the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities. As a preliminary step we also establish the time inversion property of the Rosenblatt process, as well as some technical points regarding the distribution of Z.

Potential method and projection theorems for macroscopic Hausdorff dimension

“…145] by qualifying the range of an arbitrary transient random walk. The macroscopic Hausdorff dimension was also useful for studying the large scale structure of sojourn sets associated to the Brownian motion [16], the fractional Brownian motion [3,15], and the Rosenblatt process [4].…”

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 [2,3,4,5,6,10,13,14,21,41] 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 [21]), which seeks to find a geometrically natural family of dimensions which lie between two familiar notions of dimension.…”