Related papers: Earthquake theorem for cluster algebras of finite …
A general dynamical cluster identification framework including both modeling and computation is developed. The earthquake declustering problem is studied to demonstrate how this framework applies. A stochastic model is proposed for…
We present a new method of data clustering applied to earthquake catalogs, with the goal of reconstructing the seismically active part of fault networks. We first use an original method to separate clustered events from uncorrelated…
Let $\cT$ be Teichm\"uller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured…
The number of earthquakes as a function of magnitude decays as a power law. This trend is usually justified using spring-block models, where slips with the appropriate global statistics have been numerically observed. However, prominent…
This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces…
Thurston's earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of…
Universal shape profiles in a variety of systems contain crucial information on the underlying dynamics. We develop such shape profiles for earthquakes as a stronger test of theory against observations. The earthquake analysis shows good…
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out…
An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between…
We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…
Earthquakes are one of the most devastating natural disasters that plague society. A skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional…
Using Thurston's characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a rational map…
Scaling analysis reveals striking regularities in earthquake occurrence. The time between any one earthquake and that following it is random, but it is described by the same universal-probability distribution for any spatial region and…
We present an effective method for recovering the topology of a bordered oriented surface with marked points from its cluster algebra. The information is extracted from the maximal triangulations of the surface, those that have exchange…
We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichm\"uller space $T(\mathbb{H})$ given by $\lambda\mapsto E^{\lambda}|_{S^1}$ is a homeomorphism for the…
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram…
The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws…
We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluster…
We take a novel lattice-theoretic approach to the $\tau$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice…
We investigate the sequence of great earthquakes over the past century. To examine whether the earthquake record includes temporal clustering, we identify aftershocks and remove those from the record. We focus on the recurrence time,…