arXiv subjects
Publications and source records attributed to Gabriel P. Paternain.
We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $θ$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ π+θ\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $θ$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.
We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.
We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective.
We explore the relationship between contact forms on $\mathbb S^3$ defined by Finsler metrics on $\mathbb S^2$ and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb S^2$ with curvature $K\geq 1$ and with all geodesic loops of length $>π$ is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct $J$-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on $\mathbb S^2$ with K=1 thus complementing the results obtained in \cite{HW}.
We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field $E$ has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies \[K_{w}(σ)+{1/4}|E_σ|^2<0\] for every 2-plane $σ$, where $K_{w}$ is the sectional curvature of the associated Weyl connection and $E_σ$ is the orthogonal projection of $E$ onto $σ$. Then the entropy production of any SRB measure is positive unless $E$ has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $ϕ$ determined by the motion of a particle under the influence of a magnetic field $Ω$ and a thermostat with external field ${\bf e}$. We show that if $ϕ$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and only if the external field ${\bf e}$ has a global potential $U$, $g_{1}:=e^{-2U}g$ has constant curvature and $e^{-U}Ω$ is a constant multiple of the area form of $g_1$. We also give necessary and sufficient conditions for just one of the weak foliations to be of class $C^{1,1}$ and we show that the {\it combined} effect of a thermostat and a magnetic field can produce an Anosov flow with a weak stable foliation of class $C^{\infty}$ and a weak unstable foliation which is {\it not} $C^{1,1}$. Finally we study Anosov thermostats depending quadratically on the velocity and we characterize those with smooth weak foliations. In particular, we show that quasi-fuchsian flows as defined by Ghys in \cite{Ghy1} can arise in this fashion.
We consider an optical hypersurface $Σ$ in the cotangent bundle $τ:T^*M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $Σ$ is Anosov, then a smooth 1-form $θ$ on $M$ is exact if and only $τ^*θ$ has zero integral over every closed characteristic of $Σ$. This result is derived from a related theorem about magnetic flows which generalizes our work in \cite{DP}. Other rigidity issues are also discussed.
We construct F-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if M=E#X, where E is a fiber bundle with structure group G and a fiber admitting a G-invariant metric of non-negative sectional curvature and X admits an F-structure with one trivial covering, then one can construct on M a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. $Vol_K$ (M)=0). As a corollary we prove that all the elements in the Spin cobordism group can be represented by manifolds M with $Vol_K$ (M)==0.
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,Ω)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\bar{g}$ are isometric and $a^{-1}Ω$ is, up to a sign, the area form of $g$. The characterization also implies that if a magnetic flow is Mañé critical and uniquely ergodic it must be the horocycle flow. As a by-product we also obtain results on existence of closed magnetic geodesics for almost all energy levels in the case weakly exact magnetic fields on arbitrary manifolds.
We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in such a way that the rank of the map induced in the pointed loop space homology grows exponentially. This result allows us to prove in dimensions four and five, that if M admits a metric with zero entropy then its universal covering has the rational homotopy type of a finite elliptic CW complex. We conjecture that this is the case in every dimension.
Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple of the area form.
We study the problem of existence of F-structures on compact complex surfaces, giving a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces. For instance we prove that compact Kahler surfaces of Kodaira number different from 2 have minimal entropy 0, and such a surface admits a metric with entropy 0 if and only if it is the complex projective space, a ruled surface of genus 0 or 1, a complex torus or a hyperelliptic surface. The key result we use to prove this, is a new topological obstruction to the existence of metrics with vanishing topological entropy. Finally we show that these results fit perfectly into Wall's study of geometric structures on compact complex surfaces.
We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the "non-removable intersection set" between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.
We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M if and only if M admits a metric modelled on 4 of the 8 standard 3-dimensional geometries, namely $S^3$, $S^2\times R$, $E^3$, or Nil.