arXiv research

Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.

The paper generalizes Thurston's earthquake map to cluster algebras of finite type.

Study on earthquake metric on Teichmüller space, proving properties and new completions.

Paper defines and studies measures related to earthquakes and best Lipschitz maps.

The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.

Let $\cT$ be Teichmüller 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 lamination in the Thurston boundary of $\cT$. This circle action shares some of the …

This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.

A measured laminations on the universal hyperbolic solenoid $§$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid $§$ is uniquely determined by a measured lamination on $§$; it is a leafwise earthquake with…

Extends earthquake and horocycle flows to new measures.

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

Method computes centers of Poisson and skein algebras for loops on surfaces.

The Teichmuller unipotent flow can be defined concretely on certain moduli spaces of singular flat surfaces by shearing polygonal presentations of the surfaces. Thurston's earthquake flow on moduli spaces of hyperbolic surfaces is more mysterious. Both flows have deep and important connections to other areas of mathema…

New geometric definition of Lie bracket for undirected curves.

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space $\mathcal T(X_0)$ of a noncompact Ri…

Unique geodesics selected by energy minimization in Teichmüller space.

Geodesic envelopes stay uniformly bounded in specific Teichmüller spaces.

Study earthquake deformations on a once-punctured torus.

Deep neural networks predict earthquake locations with high accuracy.

Study shows house buyers in Christchurch value earthquake risk differently based on time since 2011 quake.

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…

Deviance Voronoi residuals improve earthquake insurance risk assessment.

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ…

EQShapelets detect earthquakes with high accuracy and interpretability.

We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.

Continuity of earthquake flow map transfers Teichmüller dynamics results.

This paper was first written in 1990, but was never published. In it, the author presents a novel approach to the study of constant curvature spacetimes in 2+1 dimensions. A parameterization of flat 2+1-dimensional domains of dependence is given in terms of measured geodesic laminations. There is also an interesting re…

A machine learning surrogate model predicts earthquake-induced building responses.

New benchmark for earthquake forecasting models shows current neural point processes are not yet suitable.

The paper proves a theorem about earthquake extensions of vector fields on circles.

We prove an "Earthquake Theorem" for hyperbolic metrics with geodesic boundary on a compact surfaces $S$ with boundary: given two hyperbolic metrics with geodesic boundary on a surface with $k$ boundary components, there are $2^k$ right earthquakes transforming the first in the second. An alternative formulation arises…

Earthquake signal detection is at the core of observational seismology. A good detection algorithm should be sensitive to small and weak events with a variety of waveform shapes, robust to background noise and non-earthquake signals, and efficient for processing large data volumes. Here, we introduce the Cnn-Rnn Earthq…

Bayesian neural networks improve earthquake rupture prediction and uncertainty estimation.

Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hy…

We calculate the higher derivatives of length functions on Teichmuller space along earthquake deformations. This generalizes the cosine formula for the first derivative by Kerckhoff and Wolpert and the sine formula for second derivative by Wolpert.

Unsupervised method detects earthquakes from raw waveforms, generalizing across datasets.

Neural model outperforms ETAS in forecasting Central Apennines earthquakes.

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $π$: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``parti…

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking …

There is a small number of case studies of automatic land cover classification on the coastal area. Here, I test extraction of seagrass beds, sandy area, oyster farming rafts at Mangoku-ura Lagoon, Miyagi, Japan by comparing manual tracing, simple image segmentation, and image transformation using deep learning. The re…

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this…

Generative model synthesizes earthquake acceleration data.

Estimates Schwarzian derivative on long complex projective tubes.

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface $S$, in relation to some geometric invariants depending only on the two points in Teichmüller space of $S$ provided by Mess' parameterization - namely on two isotopy classes of hyperbolic metrics $h…

The dynamics of earthquake flow equidistributes geodesics on hyperbolic surfaces.

We find prominent similarities in the features of the time series for the (model earthquakes or) overlap of two Cantor sets when one set moves with uniform relative velocity over the other and time series of stock prices. An anticipation method for some of the crashes have been proposed here, based on these observation…

Hamenstädt gave a parametrization of the Teichmüller space of punctured surfaces such that the image under this parametrization is the interior of a polytope. In this paper, we study the Hilbert metric on the Teichmüller space of punctured surfaces based on this parametrization. We prove that every earthquake ray is an…

Spatially-aware model improves earthquake hazard assessment accuracy.

Bayesian approach models earthquake clustering with spatial mainshocks and aftershocks.