In this article, we extend, with a great deal of generality, many results regarding the Hausdorff dimension of certain dynamical Diophantine coverings and shrinking target sets associated with a conformal iterated function system (IFS) previously established under the so-called open set condition. The novelty of the result we present is that it holds regardless of any separation assumption on the underlying IFS and thus extends to a large class of IFSs the previous results obtained by Beresnevitch and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992] and by Barral and Seuret [The multifractal nature of heterogeneous sums of Dirac masses. Math. Proc. Cambridge Philos. Soc. 144(3) (2008), 707–727]. Moreover, it will be established that if S is conformal and satisfies mild separation assumptions (which are, for instance, satisfied for any self-similar IFS on $\mathbb {R}$ with algebraic parameters, no exact overlaps and similarity dimension smaller than $1$), then the classical result of Hill–Velani regarding the shrinking target problem associated with a conformal IFS satisfying the open set condition (and for which the Hausdorff measure was later computed by Allen and Barany [On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika 67 (2021), 807–839]) can be extended.

We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well.

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

The formation and the patterning of the coronary vasculature are critical to the development and pathology of the heart. Alterations in cytokine signaling and biomechanical load can alter the vascular distribution of the vessels within the heart. Changes in the physical patterning of the vasculature can have significant impacts on the relationships of the pressure-flow network and distribution of critical growth and survival factors to the tissue. Interleukin-6 (IL-6) is a pleiotropic cytokine that regulates several biological processes, including vasculogenesis. Using both immunohistological and cardioangiographic analyses, we tested the hypothesis that IL-6-loss will result in decreased vessel density, along with changes in vascular distribution. Moreover, given the impact of vascular patterning on pressure-flow and distribution mechanics, we utilized non-Euclidean geometrical fractal analysis to quantify the changes in patterning resulting from IL-6-loss. Our analyses revealed that IL-6-loss results in a decreased capillary density and increase in intercapillary distances, but does not alter vessel size or diameter. We also observed that the IL-6−/− coronary vasculature had a marked increase in fractal dimension (D value), indicating that IL-6-loss alters vascular patterning. Characterization of IL-6-loss on coronary vasculature may lend insight into the role of IL-6 in the formation and patterning of the vascular bed.

Cardiac fibroblasts are the most numerous cells in the heart and are critical in the formation and normal functioning of the organ. Cardiac fibroblasts are firmly attached to and surrounded by extracellular matrix (ECM). Mechanical forces transmitted through interaction with the ECM can result in changes of overall cellular shape, cytoskeletal organization, proliferation, and gene expression of cardiac fibroblasts. These responses may be different in the normally functioning heart, when compared with various pathological conditions, including inflammation or hypertrophy. It is apparent that cellular phenotype and physiology, in turn, are affected by multiple signal transduction pathways modulated directly by the state of polymerization of the actin cytoskeleton. Morphological changes in actin organization resulting from response to adverse conditions in fibroblasts and other cell types are basically descriptive. Some studies have approached quantifying changes in actin cytoskeletal morphology, but these have involved complex and difficult procedures. In this study, we apply image analysis and non-Euclidian geometrical fractal analysis to quantify and describe changes induced in the actin cytoskeleton of cardiac fibroblasts responding to mechanical stress. Characterization of these rapid responses of fibroblasts to mechanical stress may provide insight into the regulation of fibroblasts behavior and gene expression during heart development and disease.

Topology is an important component of the architecture of whole root systems. Unfortunately, most commonly applied indices used for characterizing topology are poorly correlated with one another and thus reflect different aspects of topology. In order to understand better how different methods of characterizing topology vary, this paper presents an exploration of several different methods for assigning order within branched root systems on the basis of (a) developmental (centrifugal) vs. functional (centripetal) ordering sequences and (b) whether orders are assigned to individual links or groups of adjacent links (segments). For each ordering system, patterns of scaling in relation to various aspects of link and segment size are explored using regression analyses. Segment-based ordering systems resulted in better fits for simple scaling relationships with size, but these patterns varied between developmental vs. functional ordering as well as the different size metrics examined. The functional (centripetal), link-based ordering system showed complex, non-linear scaling in relation to numbers of links per order. Using a simple simulation model of root growth, it is demonstrated that this method of characterizing root topology in relation to root size might be a more powerful tool for characterizing root system architecture than in the use of simple, single-index characterizations of topology.