2023/08/18 by Banaji, Amlan, Rutar, Alex, Troscheit, Sascha · 1 citation
#28A80 (Primary) #37C45 #60J80 (Secondary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2308.12975
The ϕ-Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is often closely related to "phase-transition" phenomena in sets. In this article we establish a number of key properties of the ϕ-Assouad dimensions which help to clarify their behaviour. We prove for any bounded doubling metric space F and α∈ℝ satisfying dimBF<α\leqdimA F that there is a function ϕ so that the ϕ-Assouad dimension of F is equal to α. We further show that the "upper" variant of the dimension is fully determined by the ϕ-Assouad dimension, and that homogeneous Moran sets are in a certain sense generic for these dimensions. Further, we study explicit examples of sets where the Assouad spectrum does not reach the Assouad dimension. We prove a precise formula for the ϕ-Assouad dimensions for Galton--Watson trees that correspond to a general class of stochastically self-similar sets, including Mandelbrot percolation. This result follows from two results which may be of general interest: a sharp large deviations theorem for Galton--Watson processes with bounded offspring distribution, and a Borel--Cantelli-type lemma for infinite structures in random trees. Finally, we obtain results on the ϕ-Assouad dimensions of overlapping self-similar sets and decreasing sequences with decreasing gaps.