Sparse sampling and dilation operations on Gibbs capacities, and multifractal formalism

In this article, starting from a Gibbs capacity, we build a new random capacity by applying two simple operators, the first one introducing some redundancy and the second one performing a random sampling. Depending on the values of the two parameters ruling the redundancy and the sampling, the new capacity has very different multifractal behaviors. In particular, the multifractal spectrum of the capacity may contain two to four phase transitions, and the multifractal formalism may hold only on a strict subset (sometimes, reduced to a single point) of the spectrum’s domain.

Keywords

Hausdorff dimension, random sampling, thermodynamic formalism, phase transitions, metric number theory, ubiquity theory, multifractals, large deviations

2010 Mathematics Subject Classification

28A78, 28A80, 37D35, 60F10, 60K35

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To the memory of Ka-Sing Lau, his kindness, hospitality and inspiring mathematics

Received 1 April 2023

Accepted 3 October 2023

Published 10 July 2024