Hausdorff dimension of some groups acting on the binary tree

Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T. Abert and Virag showed that there exist finitely generated (but not necessarily level-transitive) subgroups of AutT of arbitrary dimension in [0,1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then show examples of 3-generated spinal groups which have transcendental Hausdroff dimension, and exhibit a construction of 2-generated groups whose Hausdorff dimension is 1.Comment: 10 pages; full revision; simplified some proof

The three-loop Adler DD-function for N=1{\cal N}=1 SQCD regularized by dimensional reduction

The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the \overline{\mbox{DR}} scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the $D$-function in the \overline{\mbox{DR}} scheme for all renormalization constants (including the one for the electromagnetic coupling constant which appears due to the SQCD corrections). It is shown that the results do not satisfy the NSVZ-like equation relating the $D$-function to the anomalous dimension of the matter superfields. However, the NSVZ-like scheme can be constructed with the help of a properly tuned finite renormalization. It is also demonstrated that the three-loop $D$-function defined in terms of the bare couplings with the dimensional reduction does not satisfy the NSVZ-like equation for an arbitrary renormalization prescription. We also investigate a possibility to present the results in the form of the $\beta$-expansion and the scheme dependence of this expansion.Comment: 25 pages, 2 figures, 1 table, improved conclusion, version accepted for publication in JHE

Combinatorial models of expanding dynamical systems

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.Comment: The new version differs substantially from the first one. Many parts are moved to other (mostly future) papers, the main open question of the first version is solve

Adaptive Gaussian Markov Random Fields for Child Mortality Estimation

The under-5 mortality rate (U5MR), a critical health indicator, is typically estimated from household surveys in lower and middle income countries. Spatio-temporal disaggregation of household survey data can lead to highly variable estimates of U5MR, necessitating the usage of smoothing models which borrow information across space and time. The assumptions of common smoothing models may be unrealistic when certain time periods or regions are expected to have shocks in mortality relative to their neighbors, which can lead to oversmoothing of U5MR estimates. In this paper, we develop a spatial and temporal smoothing approach based on Gaussian Markov random field models which incorporate knowledge of these expected shocks in mortality. We demonstrate the potential for these models to improve upon alternatives not incorporating knowledge of expected shocks in a simulation study. We apply these models to estimate U5MR in Rwanda at the national level from 1985-2019, a time period which includes the Rwandan civil war and genocide.Comment: 59 page