Two Dimensional Water Waves in Holomorphic Coordinates

In this article we investigate spectral properties of the coupling \({H + V_\lambda}\), where \({H = -i\alpha \cdot \nabla+m\beta}\) is the free Dirac operator in \({\mathbb{R}^3}\), \({m > 0}\) and \({V_\lambda}\) is an electrostatic shell potential (which depends on a parameter \({\lambda \in \mathbb{R}}\)) located on the boundary of a smooth domain in \({\mathbb{R}^3}\). Our main result is an isoperimetric-type inequality for the admissible range of \({\lambda}\)’s for which the coupling \({H + V_\lambda}\) generates pure point spectrum in \({(-m, m)}\). That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman–Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible \({\lambda}\)’s, and we use this to relate the endpoints of the admissible range of \({\lambda}\)’s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

Positive Casimir and Central Characters of Split Real Quantum Groups

We consider the one parameter family \({\alpha \mapsto T_{\alpha}}\) (\({\alpha \in [0,1)}\)) of Pomeau-Manneville type interval maps \({T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}\) for \({x \in [0,1/2)}\) and \({T_{\alpha}(x)=2x-1}\) for \({x \in [1/2, 1]}\), with the associated absolutely continuous invariant probability measure \({\mu_{\alpha}}\). For \({\alpha \in (0,1)}\), Sarig and Gouëzel proved that the system mixes only polynomially with rate \({n^{1-1/{\alpha}}}\) (in particular, there is no spectral gap). We show that for any \({\psi \in L^{q}}\), the map \({\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}\) is differentiable on \({[0,1-1/q)}\), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For \({\alpha \ge 1/2}\) we need the \({n^{-1/{\alpha}}}\) decorrelation obtained by Gouëzel under additional conditions.     

Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels \({\Pi_{h^k}(z,w)}\) and Berezin kernels \({P_{h^k}(z,w)}\) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) where \({D(z,w)}\) is the diastasis. When the metric is only \({C^{\infty}}\) this asymptotic cannot hold for all \({(z,w)}\) since the diastasis is not even defined for all \({(z,w)}\) close to the diagonal. Our main result is that for general toric \({C^{\infty}}\) metrics, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) as long as w lies on the \({\mathbb{R}_+^m}\)-orbit of z, and for general \({(z,w)}\), \({{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}\) where \({D(z, w^*)}\) is the diastasis between z and the translate of w by \({(S^1)^m}\) to the \({\mathbb{R}_+^m}\) orbit of z. These results are complementary to Mike Christ’s negative results showing that \({P_{h^k}(z,w)}\) does not have off-diagonal exponential decay at “speed” k if \({(z,w)}\) lies on the same \({(S^1)^m}\)-orbit.     

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations \({\varGamma }^+ \times {\varGamma }^-\) over \({\mathbb {R}}^d\). Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator \(L^+(\gamma ^-) + \frac{1}{\varepsilon }L^-\), \(\varepsilon > 0\). Here \(L^-\) describes the environment process on \({\varGamma }^-\) and \(L^+(\gamma ^-)\) describes the system process on \({\varGamma }^+\), where \(\gamma ^-\) indicates that the corresponding birth-and-death rates depend on another locally finite configuration \(\gamma ^- \in {\varGamma }^-\). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states \(\mu _t^{\varepsilon }\) on \({\varGamma }^+ \times {\varGamma }^-\). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let \(\mu _{\mathrm {inv}}\) be the invariant measure for the environment process on \({\varGamma }^-\). In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of \(\mu _t^{\varepsilon }\) onto \({\varGamma }^+\) converges weakly to an evolution of states on \({\varGamma }^+\) associated with the averaged Markov birth-and-death operator \({\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)\).     

Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group \({\exp(-\mathrm{i} t H_0)}\) of the discrete Laguerre operator H₀ (i.e., the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression, we show that the norm of the evolution group acting from \({\ell^1}\) to \({\ell^\infty}\) is given by \({(1+t^2)^{-1/2}}\).     

On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I

We study the determinant \({\det(I-\gamma K_s), 0 < \gamma < 1}\) , of the integrable Fredholm operator K _s acting on the interval (?1, 1) with kernel \({K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}\) . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature \({\beta=2}\) , in the presence of an external potential \({v=-\frac{1}{2}\ln(1-\gamma)}\) supported on an interval of length \({\frac{2s}{\pi}}\) . We evaluate, in particular, the double scaling limit of \({\det(I-\gamma K_s)}\) as \({s\rightarrow\infty}\) and \({\gamma\uparrow 1}\) , in the region \({0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}\) , for any fixed \({0 < \delta < 1}\) . This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995).     

The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data

The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data \({\mathcal{D}{\rm Hg}}\) fits into a family \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) , where n ≥  0 and \({\omega\in \mathbb{Z}_{2n+1}}\) . We show \({\mathcal{D}^0 {\rm Hg}_{2n+1}}\) is related to the subfactors Izumi hypothetically associates to the cyclic groups \({\mathbb{Z}_{2n+1}}\) . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \({\mathbb{Z}_7, \mathbb{Z}_9}\) and \({\mathbb{Z}_3^2}\) , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \({\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}\) (previously, Izumi had shown uniqueness for \({\mathbb{Z}_3}\) and \({\mathbb{Z}_5}\)), and we identify their modular data. We explain how \({\mathcal{D}{\rm Hg}}\) (more generally \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\)) is a graft of the quantum double \({\mathcal{D} Sym(3)}\) (resp. the twisted double \({\mathcal{D}^\omega D_{2n+1}}\)) by affine so(13) (resp. so\({(4n^2+4n+5)}\)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.     

Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment

It is shown that the deterministic infinite trigonometric products

$$\begin{aligned} \prod _{n\in \mathbb {N}}\left[ 1- p +p\cos \left( \textstyle n^{-s}_{_{}}t\right) \right] =: {\text{ Cl }_{p;s}^{}}(t) \end{aligned}$$

with parameters \( p\in (0,1]\ \& \ s>\frac{1}{2}\), and variable \(t\in \mathbb {R}\), are inverse Fourier transforms of the probability distributions for certain random series \(\Omega _{p}^\zeta (s)\) taking values in the real \(\omega \) line; i.e. the \({\text{ Cl }_{p;s}^{}}(t)\) are characteristic functions of the \(\Omega _{p}^\zeta (s)\). The special case \(p=1=s\) yields the familiar random harmonic series, while in general \(\Omega _{p}^\zeta (s)\) is a “random Riemann-\(\zeta \) function,” a notion which will be explained and illustrated—and connected to the Riemann hypothesis. It will be shown that \(\Omega _{p}^\zeta (s)\) is a very regular random variable, having a probability density function (PDF) on the \(\omega \) line which is a Schwartz function. More precisely, an elementary proof is given that there exists some \(K_{p;s}^{}>0\), and a function \(F_{p;s}^{}(|t|)\) bounded by \(|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})\), and \(C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi \), such that

$$\begin{aligned} \forall \,t\in \mathbb {R}:\quad {\text{ Cl }_{p;s}^{}}(t) = \exp \bigl ({- C_{p;s}^{} \,|t|^{1/s}\bigr )F_{p;s}^{}(|t|)}; \end{aligned}$$

the regularity of \(\Omega _{p}^\zeta (s)\) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that \(\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) \) for some \(C>0\). Graphical evidence suggests that \({\text{ Cl }_{{{1}/{3}};2}^{}}(t)\) is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of \({\text{ Cl }_{{{1}/{3}};2}^{}}\)), and illustrated by random sampling of the Riemann-\(\zeta \) walks, whose branching rules allow the build-up of fractal-like structures.

     

Crossover to the Stochastic Burgers Equation for the WASEP with a Slow Bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter \({\rho \in (0,1)}\). The rate of passage of particles to the right (resp. left) is \({\frac{1}{2} + \frac{a}{2n^{\gamma}}}\) (resp. \({\frac{1}{2} - \frac{a}{2n^{\gamma}}}\)) except at the bond of vertices \({\{-1,0\}}\) where the rate to the right (resp. left) is given by \({\frac{\alpha}{2n^\beta} + \frac{a}{2n^{\gamma}}}\) (resp. \({\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\gamma}}}\)). Above, \({\alpha > 0}\), \({\gamma \geq \beta \geq 0}\), \({a\geq 0}\). For \({\beta < 1}\), we show that the limit density fluctuation field is an Ornstein–Uhlenbeck process defined on the Schwartz space if \({\gamma > \frac{1}{2}}\), while for \({\gamma = \frac{1}{2}}\) it is an energy solution of the stochastic Burgers equation. For \({\gamma \geq \beta =1}\), it is an Ornstein–Uhlenbeck process associated to the heat equation with Robin’s boundary conditions. For \({\gamma \geq \beta > 1}\), the limit density fluctuation field is an Ornstein–Uhlenbeck process associated to the heat equation with Neumann’s boundary conditions.     

On Essential Self-Adjointness for Magnetic Schrödinger and Pauli Operators on the Unit Disc in $${\mathbb {R}^2}$$

We study the question of magnetic confinement of quantum particles on the unit disk \({\mathbb {D}}\) in \({\mathbb {R}^2}\) , i.e. we wish to achieve confinement solely by means of the growth of the magnetic field \({B(\vec x)}\) near the boundary of the disk. In the spinless case, we show that \({B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}}\) , for \({|\vec x|}\) close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants \({\frac{\sqrt 3}{2}}\) and \({-\frac{1}{\sqrt 3}}\) are optimal. This answers, in this context, an open question from Colin de Verdière and Truc (Ann Inst Fourier 2011, Preprint, arXiv:0903.0803v3). We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of {GL(2,\mathbb{Z})}

Spectral measures for fundamental representations of the rank two Lie groups SU(3), Sp(2) and G₂ have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus \({\mathbb{T}^2}\) and are invariant under an action of the corresponding Weyl group, which is a subgroup of \({GL(2,\mathbb{Z})}\). Here we consider spectral measures invariant under an action of the other finite subgroups of \({GL(2,\mathbb{Z})}\). These spectral measures are all associated with fundamental representations of other rank two Lie groups, namely \({\mathbb{T}^2=U(1) \times U(1)}\), \({U(1) \times SU(2)}\), U(2), \({SU(2) \times SU(2)}\), SO(4) and PSU(3).     

On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥?2. We prove that there exists a C²-residual subset \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if the closure of the set of periodic orbits of \({\varphi ^{t}_{g}}\) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C²-residual \(\mathscr{R}\) such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if \({\varphi ^{t}_{g}}\) is an Anosov flow.