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Mathematics > Probability

arXiv:2203.15315 (math)
[Submitted on 29 Mar 2022 (v1), last revised 24 Oct 2022 (this version, v2)]

Title:Box-counting dimension in one-dimensional random geometry of multiplicative cascades

Authors:Kenneth J. Falconer, Sascha Troscheit
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Abstract:We investigate the box-counting dimension of the image of a set $E \subset \mathbb{R}$ under a random multiplicative cascade function $f$. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for the box-counting dimension. However, we show that this is far from true in general, and we compute explicitly a formula of a very different nature that gives the almost sure box-counting dimension of the random image $f(E)$ when the set $E$ comprises a convergent sequence. In particular, the box-counting dimension of $f(E)$ depends more subtly on $E$ than just on its dimensions. We also obtain lower and upper bounds for the box-counting dimension of the random images for general sets $E$.
Comments: 26 pages, 4 figures
Subjects: Probability (math.PR); Dynamical Systems (math.DS); Metric Geometry (math.MG)
Cite as: arXiv:2203.15315 [math.PR]
  (or arXiv:2203.15315v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.2203.15315
Related DOI: https://doi.org/10.1007/s00220-022-04558-9

Submission history

From: Sascha Troscheit [view email]
[v1] Tue, 29 Mar 2022 07:57:48 UTC (100 KB)
[v2] Mon, 24 Oct 2022 14:17:18 UTC (126 KB)
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