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This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's…
City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature…
The purpose of the paper is a general analysis of path space measures. Our focus is a certain path space analysis on generalized Bratteli diagrams. We use this in a systematic study of systems of self-similar measures (the term ``IFS…
Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized…
In this paper we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We show that $p$-adic self-similar sets are $p$-adic path set fractals, and that the converse is not necessarily true. For $p$-adic self-similar…
We consider four prototypes of variational problems and prove the existence of fractal minimizers through the direct method in the calculus of variations. By design these minimizers are H\"older curves or H\"older parametrizations of…
Fractal geometry, defined by self-similar patterns across scales, is crucial for understanding natural structures. This work addresses the fractal inverse problem, which involves extracting fractal codes from images to explain these…
We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order…
A class of simplified measures is constructed to capture the key features of generic spatio-temporally chaotic systems. A combined analytical and numerical investigation allows us to extablish the scaling beahviour of the fractal dimension…
We give a generalization of Lagarias' formula for diffraction by ideal crystals, and we apply it to the lattice case, in preparation for addressing the problem of quasicrystals and complex dimensions posed by Lapidus and van Frankenhuijsen…
We forge connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric…
For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $\br^d$, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$…
Much of the structure in metric spaces that allows for the creation of fractals exists in more generalized non-metrizable spaces. In particular the same theorems regarding the behavior of compact sets can be proven in the more general…
We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are…
We establish a correspondence between Pavlovian conditioning processes and fractals. The association strength at a training trial corresponds to a point in a disconnected set at a given iteration level. In this way, one can represent a…
We establish a general analytic framework for determining the AF-martingale dimension of diffusion processes associated with strongly local regular Dirichlet forms on metric measure spaces. While previous approaches typically relied on…
By appropriate choices of elements in the underlying iterated function system, methodology of fractal interpolation entitles one to associate a family of continuous self-referential functions with a prescribed real-valued continuous…
This paper deals with iteration stable (STIT) tessellations, and, more generally, with a certain class of tessellations that are infinitely divisible with respect to iteration. They form a new, rich and flexible class of spatio-temporal…
Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions…
We derive explicit integrability conditions for stochastic integrals taken over time and space driven by a random measure. Our main tool is a canonical decomposition of a random measure which extends the results from the purely temporal…