Related papers: Slicing Theorems and rigidity phenomena for self a…

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We…

We consider a random self-affine carpet $F$ based on an $n\times m$ subdivision of rectangles and a probability $0<p<1$. Starting by dividing $[0,1]^2$ into an $n\times m$ grid of rectangles and selecting these independently with…

We show that Thurston's skinning maps of Teichmuller space have finite fibers. The proof centers around a study of two subvarieties of the SL_2(C) character variety of a surface, one associated to complex projective structures and the other…

Let $s\in (0,1)$, and let $F\subset \mathbb{R}$ be a self similar set such that $0 < \dim_H F \leq s$ . We prove that there exists $\delta= \delta(s) >0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$…

Let $E, F\subset {\Bbb R}^d$ be two self-similar sets, and suppose that $F$ can be affinely embedded into $E$. Under the assumption that $E$ is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between…

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show…

Let $E$ be a Lalley-Gatzouras carpet determined by a set of contractive affine mappings $\{f_{ij}\}_{(i,j)\in G}$. We study the asymptotics of quantization error for the self-affine measures $\mu$ on $E$. We prove that the upper and lower…

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a precise formula for…

The connected part of a metric space $E$ is defined to be the union of non-trivial connected components of $E$. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of $E$ either coincides…

Given an entire transcendental function f with a non-completely invariant Baker domain, we define a Baker lamination on geodesics to study the divergence and convergence of a pinching process of curves in U. If the boundary of some curve in…

Given a smooth affine curve X over a field k of positive characteristic, and an overconvergent F-isocrystal on X, we prove after replacing k by a finite purely inseparable extension, there exists a finite separable cover of X, the pullback…

We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its…

Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded…

We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain $\Omega$ on the Riemann sphere with conformally non-removable boundary. Here rigidity means that every conformal map from $\Omega$…

We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As…

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_\epsilon$ be a ball in $U$ of…

Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The universal deformation…

We show that any self-conformal measure $\mu$ on $\mathbb{R}$ is uniformly scaling and generates an ergodic fractal distribution. This generalizes existing results by removing the need for any separation condition. We also obtain…

Let $\mu$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there…

This article concerns properties of mixed $\ell$-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of…