A note on the fractional perimeter and interpolation

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{\alpha, 1}$ of order $0 < \alpha < 1$

Specialization of Difference Equations and High Frobenius Powers

We study valued fields equipped with an automorphism $\sigma$ which is locally infinitely contracting in the sense that $\alpha\ll\sigma\alpha$ for all $0<\alpha\in\Gamma$. We show that various notions of valuation theory, such as Henselian and strictly Henselian hulls, admit meaningful transformal analogues. We prove canonical amalgamation results, and exhibit the way that transformal wild ramification is controlled by torsors over generalized vector groups. Model theoretically, we determine the model companion: it is decidable, admits a simple axiomatization, and enjoys elimination of quantifiers up to algebraically bounded quantifiers. The model companion is shown to agree with the limit theory of the Frobenius action on an algebraically closed and nontrivially valued field. This opens the way to a motivic intersection theory for difference varieties that was previously available only in characteristic zero. As a first consequence, the class of algebraically closed valued fields equipped with a distinguished Frobenius $x\mapsto x^{q}$ is decidable, uniformly in $q$.Comment: identical to v1 apart from slight modifications in abstrac

Alcove path model for B(∞)B(\infty)

We construct a model for $B(\infty)$ using the alcove path model of Lenart and Postnikov. We show that the continuous limit of our model recovers a dual version of the Littelmann path model for $B(\infty)$ given by Li and Zhang. Furthermore, we consider the dual version of the alcove path model and obtain analogous results for the dual model, where the continuous limit gives the Li and Zhang model.Comment: 19 pages, 7 figures; improvements from comments, added more figure

Local connectivity of the Mandelbrot set at some satellite parameters of bounded type

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family

Harmonic analysis on the boundary of hyperbolic groups

In this paper we show that a Möbius-structure $\mathcal{M}$ of dimension $Q$ has a minimal Ahlfors-David constant.This shows that a Möbius space is uniformly $Q$-Ahlfors-David regular.In summary, many classical theorems of harmonic analysis on $\mathbb{R}^n$ admit a Möbius-invariant formulation in the context of Möbius-geometry.We use this observation to show that the Knapp-Stein operator$$(I_d^\alpha u_d)(x) = \int \frac{u_d(y)}{d(x,y)^{Q - \alpha}} \, d\mu_d(y), \quad\quad (\, 0 < \alpha < \frac{Q}{2}\,)$$is a continuous operator on the weighted $L^2$-space $L^2((\frac{d'}{d})^{\alpha} d\mu_d)$, with a norm independent of $d$ and $d'$.From here we construct a Sobolev space $\mathcal{H}^{-\alpha}_d$ on $s$-densities for a given $s$ as a function of $\alpha$.We would like to say that the construction is topologically independent of the metric $d$.In this paper we prove that the norms on a large class of functions are comparable.The work is inspired by a paper by Astengo, Cowling, and Di Blasio, who construct uniformly bounded representations for simple Lie groups of rank $1$.We formulate the problem in a much more general framework of groups acting on Möbius structures.In particular, all hyperbolic groups