Cohomological Equation Research Articles

Quasi-quantum groups from strings

Motivated by string theory on the orbifold M/G in presence of a Kalb-Ramond field strength H, we define the operators that lift the group action to the twisted sectors. These operators turn out to generate the quasi-quantum group Dω[G], introduced in the context of conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche [1], with ω a 3-cocycle determined by a series of cohomological equations in a tricomplex combining de Rham, Cech and group cohomologies. We further illustrate some properties of the quasi-quantum group from a string theoretical point of view. This work is based on [2], from which a full-fledged treatment may be extracted.

On the cohomological equation of magnetic flows

We consider a magnetic flow without conjugate points on a closed manifold M with generating vector field Gμ. Let h ∈ C∞(M) and let θ be a smooth 1-form on M . We show that the cohomological equation Gμ(u) = h ◦ π + θ has a solution u ∈ C∞(SM) only if h = 0 and θ is closed. This result was proved in [10] under the assumption that the flow of Gμ is Anosov.

Axiomatic theory of divergent series and cohomological equations

A general theory of summation of divergent series based on the Hardy�Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Kolmogorov.

Analytical Moduli for Unfoldings of Saddle-Node Vector Fields

In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the vector field with the saddle-node. The point of view is to compare the with a model family via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences for solutions of the cohomological equation.

An additive cohomological equation and typical behavior of Birkhoff sums over a translation of the multidimensional torus

For a periodic function f with a given decrease of the moduli of its Fourier coefficients, we analyze the solvability of the equation \(w(T_\alpha x) - w(x) = f(x) - \smallint _{\mathbb{T}^d } f(t) dt\) and the asymptotic behavior of the Birkhoff sums Σs=0n−1f(Tαsx) for almost every α. The results obtained are applied to the study of ergodic properties of a cylindrical cascade and of a special flow on the torus.

On the symmetries of special holonomy sigma models

In addition to superconformal symmetry, (1,1) supersymmetric two-dimensional sigma models on special holonomy manifolds have extra symmetries that are in one-to-one correspondence with the covariantly constant forms on these manifolds. The superconformal algebras extended by these symmetries close as W-algebras, i.e. they have field-dependent structure functions. It is shown that it is not possible to write down cohomological equations for potential quantum anomalies when the structure functions are field-dependent. In order to do this it is necessary to linearise the algebras by treating composite currents as generators of additional symmetries. It is shown that all cases can be linearised in a finite number of steps, except for G_2 and SU(3). Additional problems in the quantisation procedure are briefly discussed.

On the cohomological equation for nilflows

Let $X$ be a vector field on a compact connected manifold $M$. An important question in dynamical systems is to know when a function $g: M\to \mathbb{R}$ is a coboundary for the flow generated by $X$, i.e., when there exists a function $f: M\to \mathbb{R}$ such that $Xf=g$. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions $D_n$ such that any sufficiently smooth function $g$ is a coboundary iff it belongs to the kernel of all the distributions $D_n$.

Livšic regularity for Markov systems

We prove measurable Livsic theorems for dynamical systems modelled by Markov towers. Our reguarlity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We consider both Holder cocycles and cocyles with singularities of prescribed order.

The cohomological equation for Roth-type interval exchange maps

We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \Phi belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund’s theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same. The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to T T . More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure. In the appendices we construct concrete examples of Roth-type i.e.m.’s and we show how the growth rate condition alone does not imply unique ergodicity.

On the additive cohomological equation and time change for a linear flow on the torus with a Diophantine frequency vector

For a 1-periodic function of finite smoothness and a Diophantine vector the solubility problem is studied for the additive cohomological equation on the torus where is the shift of the torus by the vector and is an unknown measurable function. Necessary and sufficient conditions are obtained for the conjugacy of a linear flow on the -torus to the reparametrized flow where is a positive 1-periodic function of finite smoothness.

Invariant distributions and time averages for horocycle flows

There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.

On absolutely continuous weakly mixing cocycles over irrational rotations

A weakly mixing cocycle over a rotation is a measurable function , where , such that the equation for almost all (*) has no measurable solutions for any and , . If the irrational number has bounded convergents in its continued fraction expansion and a function increases more slowly than , then it is proved that there exists a weakly mixing cocycle of the form , where belongs to the class . In addition, it is shown that equation (*) (and also the corresponding additive cohomological equation) is soluble for .

On the cohomological equation for interval exchange maps

We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation Ψ− Ψ∘ T= Φ has a bounded solution Ψ provided that the datum Φ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The class of interval exchange maps is characterized in terms of a diophantine condition of “Roth type” imposed to an acceleration of the Rauzy–Veech–Zorich continued fraction expansion associated to T. To cite this article: S. Marmi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On solvability of the cohomology equation in function spaces

Let T be an endomorphism of a probability measure space ( ;A; ), and f be a real-valued measurable function on . We consider the cohomology equation f = h T h. Conditions for the existence of real-valued measurable solutions h in some function spaces are deduced. The results obtained generalize and improve a recent result of Alonso, Hong and Obaya.

Quasianalytic monogenic solutions of a cohomological equation

Introduction Monogenic properties of the solutions of the cohomological equation Carleman classes at diophantine points Resummation at resonances and constant-type points Conclusions and applications Appendix Bibliography.

A note on a generalized cohomology equation

We give a necessary and sufficient condition for the solvability of a generalized cohomology equation, for an ergodic endomorphism of a probability measure space, in the space of measurable complex functions. This generalizes a result obtained in [7].

ON SOME SIDON SEQUENCES

We study sequences (f ◦ φn), where φ is an ergodic continuous endomorphism of a compact nontrivial Abelian group G with the probability Haar measure μ and f has an absolutely summable Fourier series. We prove that if (f ◦ φn) is a Sidon sequence (i.e. the uniform norm and l1-norm are equivalent on the linear space spanned by f ◦ φn, n≥ 0), then the spectral density of f-μ(f) with respect to the dynamical system (G, μ, φ) is positive. The converse of the theorem holds under an additional assumption on f. For f being a trigonometric polynomial, this characterization is complete and can be reformulated in terms of unsolvability of a generalized cohomology equation for f with respect to the dynamical system (G, μ, φ). It turns out that the set of those f∈ C(G) for which (f ◦ φn) is a Sidon sequence is open and dense in C(G).

A Banach algebra version of the Livsic theorem

The goal of this paper is to study Banach algebra valued cocycles over Anosov actions. The study of cohomological equations over Anosov diffeomorphisms and flows was started in two influential papers by Livsic ([L1], [L2]), and experienced later a tremendous development. This paper is a continuation of [NT1] and [NT2]. We show here that the techniques used to study cocycles with values in Lie groups, and with values in diffeomorphism groups, can be adapted to Banach algebra valued cocycles. Results of this nature are necessary for the study of extensions of Anosov group actions on infinite dimensional manifolds.

Solutions of the Cohomological Equation for Area-Preserving Flows on Compact Surfaces of Higher Genus

Solutions of the Cohomological Equation for Area-Preserving Flows on Compact Surfaces of Higher Genus

Analytic regularity of solutions of Livsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems

We study Livsic's problem of finding $\phi$ satisfying $X\phi=\eta$, where $\eta$ is a given function and $X$ is a given Anosov vector field. We show that, if $\phi$ is a continuous solution and $X,\eta$ are analytic, then $\phi$ is analytic. We use the previous result to show that if two low-dimensional Anosov systems are topologically conjugate and the Lyapunov exponents at corresponding periodic points agree, the conjugacy is analytic. Analogous results hold for diffeomorphisms.