Filtration and splitting of the Hodge bundle on the nonvarying strata of quadratic differentials

By the results of G. Forni and of R. Treviño, the Lyapunov spectrum of the Hodge bundle over the Teichmüller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those \mathrm{PSL}(2,\mathbb R) -invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss–Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmüller curve, for which the real Hodge bundle does not contain any \mathrm{PSL}(2,\mathbb R) -invariant, subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any \mathrm{PSL}(2,\mathbb R) -invariant probability measure.

A central goal of population ecology is to understand and predict fluctuations in population numbers. Until recently, much of the debate focused on the issue of population regulation by density‐dependent factors. In this paper, I describe an approach to nonlinear modeling of time‐series data that is designed to go beyond this question by investigating the possibility of complex population dynamics, characterized by lags in regulation and periodic or chaotic oscillations. The questions motivating this approach are: what are relative contributions of endogenous vs. exogenous components of dynamics? Is the irregular component in fluctuations entirely due to exogenous noise, or do nonlinearities contribute to it, too? I describe the philosophy and the technical details of the nonlinear modeling approach, and then apply it to a collection of time‐series data on vole population fluctuations in northern Europe. The results suggest that population dynamics of European voles undergo a latitudinal shift from stability to chaos. Dynamics in northern Fennoscandia are characterized by positive Lyapunov exponent estimates, and a high degree of short‐term (one year ahead) predictability, suggesting a strong endogenous component. In more southerly populations estimated Lyapunov exponents are negative, and there is no one‐step ahead predictability, suggesting that fluctuations are driven by exogenous factors.

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

Comment on “Metrics to describe the dynamical evolution of atmospheric moisture: Intercomparison of model (NARR) and observations (ISCCP)” by Kun Tao and Ana P. Barros

The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum of the Lyapunov exponents associated with the Teichm\uller flow on the Hodge bundle.

This paper deals with the problem of the discrimination between stable and unstable time series. One criterion for the seperation is given by the size of the Lyapunov exponent, which was originally defined for deterministic systems. However, this paper will show, that the Lyapunov exponent can also be analyzed and used for ergodic stochastic time series. Experimantal results illustrate the classification by the Lyapunov exponent. Although the Lyapunov exponent is a discriminatory parameter of the asymptotic behavior and can be interpreted as a parameter of the asymptotic distribution in the stochastic case, it has to be estimated from a given time series, where the process might still be in the transient state. Experimental results will show that in special cases the estimation leads to misclassifications of the time series and the underlying process due to the uncertainty of estimators for the Lyapunov exponent.

We construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a ‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k‐Hodge bundle determined by Gheorghita–Tarasca and by Korotkin–Sauvaget–Zograf.

We prove that the closed orbit of the Eierlegende Wollmilchsau is the only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>SL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -orbit closure in genus three with a zero Lyapunov exponent in its Kontsevich–Zorich spectrum. The result recovers previous partial results in this direction by Bainbridge–Habegger–Möller and the first named author. The main new contribution is the identification of the differentials in the Hodge bundle corresponding to the Forni subspace in terms of the degenerations of the surface. We use this description of the differentials in the Forni subspace to evaluate them on absolute homology curves and apply the jump problem from the work of Hu and the third named author to the differentials near the boundary of the orbit closure. This results in a simple geometric criterion that excludes the existence of a Forni subspace.

Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we\nshow that the direct image of an adjoint semi-ample line bundle by a projective\nsubmersion has a continuous metric with Griffiths semi-positive curvature. This\nshows that for every holomorphic semi-ample vector bundle $E$ on a complex\nmanifold, and every positive integer $k$, the vector bundle $S^kE\\otimes\\det E$\nhas a continuous metric with Griffiths semi-positive curvature. If $E$ is ample\non a projective manifold, the metric can be made smooth and Griffiths positive.\n

Lyapunov exponents, defined as exponential divergent or convergent rate of initially infinitely close solution trajectories, have been widely used for diagnosing chaotic systems, as well for stability analysis of nonlinear systems. Although calculated from the evolution of disturbance vectors associated with the flow, Lyapunov exponents are not associated with any specific directions, and such evolutions are driven by the dynamics in all directions in the state space. It is desirable to explore the asymptotic behaviors of the dynamic systems along certain specific directions and the specific dynamics driving such behaviors. In this paper, the Lyapunov exponents are modified. The modified Lyapunov exponents can indicate the exponential divergent or convergent rates in certain directions, which are driven by the dynamics in the same directions. The existence and the invariance to the initial conditions of the proposed modified exponents are proven mathematically. The algorithm for calculating the modified Lyapunov exponents from mathematical models is also developed. A wide range of case studies, from classical nonlinear dynamic systems to engineering systems, are presented to demonstrate the proposed modified Lyapunov exponents, and the indications of the modified exponents are also discussed. The proposed modified Lyapunov exponents can reveal additional insights into the system dynamics to the conventional Lyapunov exponents. Such information can be instrumental for stability control design.

We define the Weierstrass filtration for Teichmüller curves andconstruct the Harder-Narasimhan filtration of the Hodge bundle of aTeichmüller curve in hyperelliptic loci and low-genus nonvaryingstrata. As a result we obtain the sum of Lyapunov exponents ofTeichmüller curves in these strata.

Lyapunov exponents constitute a class of parameters which describe the asymptotic behavior of a large class of dynamical processes in chemistry and physics. This paper gives a variational characterization of the largest Lyapunov exponent for a class of models of chemical kinetics described by products of random nonnegative matrices. We show that for this class of models the largest Lyapunov exponent satisfies an extremal principle formally identical to the minimization of the quenched free energy in random spin models. This extremal principle, which yields a computable expression for the Lyapunov exponent, implies that fluctuations in the Lyapunov exponent, due to a certain class of perturbations in the matrix elements, are determined by a macroscopic parameter which is the analog of the mean energy in random spin systems. These results characterize a class of random models in chemical kinetics that are thermodynamically stable in the sense that they possess an asymptotic limit in which analogs of the laws of equilibrium thermodynamics hold.

Lyapunov exponents characterize the asymptotic behavior of long matrix products. In this work we introduce a new technique that yields quantitative lower bounds on the top Lyapunov exponent in terms of an efficiently computable matrix sum in ergodic situations. Our approach rests on two results from matrix analysis—the n-matrix extension of the Golden–Thompson inequality and an effective version of the Avalanche Principle. While applications of this method are currently restricted to uniformly hyperbolic cocycles, we include specific examples of ergodic Schrödinger cocycles of polymer type for which outside of the spectrum our bounds are substantially stronger than the standard Combes–Thomas estimates. We also show that these techniques yield short proofs of quantitative stability results for the top Lyapunov exponent which are known from more dynamical approaches. We also discuss the problem of finding stable bounds on the Lyapunov exponent for almost-commuting matrices.

Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.