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Register a new userA criterion for being a Teichmuller curve
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Authors
Elise Goujard
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Teichmuller curves play an important role in the study of dynamics in polygonal billiards. In this article, we provide a criterion similar to the original Mollers criterion, to detect whether a complex curve, embedded in the moduli space of Riemann surfaces and endowed with a line subbundle of the Hodge bundle, is a Teichmuller curve, and give a dynamical proof of this criterion.
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In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmueller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmueller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmueller curves. Finally, we extend the discussion in this note to show the simplicity of Lyapunov exponents coming from (high weight) variations of Hodge structures associated to mirror quintic Calabi-Yau threefolds.
Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov cite{0007} introduced the matrix $A_{alpha}(G)=alpha D(G)+(1-alpha)A(G)$ for $alphain [0, 1].$ The $A_alpha$-spectrum of a graph $G$ consists of all the eigenvalues (including the multiplicities) of $A_alpha(G).$ A graph $G$ is said to be determined by the generalized $A_{alpha}$-spectrum (or, DGA$_alpha$S for short) if whenever $H$ is a graph such that $H$ and $G$ share the same $A_{alpha}$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, when $alpha$ is rational, we present a simple arithmetic condition for a graph being DGA$_alpha$S. More precisely, put $A_{c_alpha}:={c_alpha}A_alpha(G),$ here ${c_alpha}$ is the smallest positive integer such that $A_{c_alpha}$ is an integral matrix. Let $tilde{W}_{{alpha}}(G)=left[{bf 1},frac{A_{c_alpha}{bf 1}}{c_alpha},ldots, frac{A_{c_alpha}^{n-1}{bf 1}}{c_alpha}right]$, where ${bf 1}$ denotes the all-ones vector. We prove that if $frac{det tilde{W}_{{alpha}}(G)}{2^{lfloorfrac{n}{2}rfloor}}$ is an odd and square-free integer and the rank of $tilde{W}_{{alpha}}(G)$ is full over $mathbb{F}_p$ for each odd prime divisor $p$ of $c_alpha$, then $G$ is DGA$_alpha$S except for even $n$ and odd $c_alpha,(geqslant 3)$. By our obtained results in this paper we may deduce the main results in cite{0005} and cite{0002}.
The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmuller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmuller structure. Moreover, this Teichmuller structure is a complete structure and is the completion of the space of smooth potentials under this Teichmuller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.
We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.
We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1.$ We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.
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