Related papers: Hausdorff dimensions for graph-directed measures d…
This paper investigates a broad class of non-Gaussian measures, $ \mu_\Psi$, associated with a family of generalized Wright functions, $_m\Psi_q$. First, we study these measures in Euclidean spaces $\mathbb{R}^d$, then define them in an…
Dimensions of level sets of generic continuous functions and generic H\"older functions defined on a fractal $F$ encode information about the geometry, ``the thickness" of $F$. While in the continuous case this quantity is related to a…
The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of Outer Space of the corresponding irreducible outer automorphism of a free group.…
We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for…
We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…
We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been…
We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and…
We study the asymptotic quantization error of order $r$ for Markov-type measures $\mu$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $\mu$ exists and determine its exact value $s_{r}$ in…
Measures on Fra\"iss\'e classes are a key input in the Harman--Snowden (2022) construction of tensor categories. Treelike Fra\"iss\'e classes provide a particularly tractable source of examples. In this paper, we complete the classification…
We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…
In this article, we investigate the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choice of continuous functions on a rectangular region. We estimate bounds for the box dimension and the…
We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then…
We investigate path-, ray- and branch spaces of trees, certain topological spaces naturally associated with order theoretic trees, and provide topological characterisations for these spaces in terms of the existence of certain kinds of…
We work on a Galton--Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$-th level of the tree, as $n$ goes to infinity, by a mostly analytic…
In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower…
We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE…
The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses…
In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n we show that this p harmonic measure is…
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its…
Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric…