arXiv subjects

Francisco Torres de Lizaur

Publications and source records attributed to Francisco Torres de Lizaur.

13 recordsLinked to original sources

arXiv2025

On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

The helicity, or asymptotic linking number, is a functional of exact volume-preserving vector fields on 3-manifolds, invariant under volume-preserving transformations. It is known to exhibit remarkable uniqueness properties: many invariant functionals reduce to functions of helicity. We examine how severely this uniqueness can fail. On integral homology spheres with the $C^{1}$-topology, the failure is extreme: for every $C^{1}$-open set of nonvanishing exact fields of fixed helicity, some other global dynamical invariant is continuous and non-constant in that set; the Ruelle invariant if some field is non-Anosov, and topological entropy otherwise. In particular, on the three-sphere, the Ruelle invariant is everywhere independent of helicity. This implies, in a very strong sense, a negative answer to a question of Arnold and Khesin on the somewhere density of coadjoint orbits of 3D volume-preserving diffeomorphisms, when considered for the $C^1$-topology. On arbitrary three-manifolds, we also answer the question in the negative using local rather than global invariants.

math.DS↗

arXiv2024

Contact type solutions and non-mixing of the 3D Euler equations

We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the $C^{1,s}$-topology, producing countably many disjoint invariant open sets.

math.DS↗

arXiv2022

Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher

Building on the work of Crouseilles and Faou on the 2D case, we construct $C^\infty$ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer $N\geq 1$ we prove that any $L^q$ initial stream function can be approximated in $L^q$ (strongly when $1\leq q< \infty$ and weak-* when $q=\infty$) by smooth initial data whose solutions are dense on $N$-dimensional tori.

math.AP↗

arXiv2022

Distribution symmetry of toral eigenfunctions

In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we provide a counterexample for higher dimensional tori, which relies on a computer assisted argument. Moreover we prove a theorem on the distribution symmetry of a certain class of trigonometric polynomials that might be of independent interest.

math.CA↗

arXiv2022

Sign equidistribution of Legendre polynomials

We prove {\em sign equidistribution} of Legendre polynomials: the ratio between the lengths of the regions in the interval $[-1, 1]$ where the Legendre polynomial assumes positive versus negative values, converges to one as the degree grows. The proof method also has application to the symmetry conjecture for a basis of eigenfunctions in the sphere.

math.CA↗

arXiv2022

Limiting measures and energy growth for sequences of solutions to Taubes's Seiberg-Witten equations

We consider sequences of solutions $(ψ_n,A_n)_{n=1}^\infty$ to Taubes's modified Seiberg-Witten equations, associated with a fixed volume-preserving vector field $X$ on a 3-manifold and corresponding to arbitrarily large values of the strength parameter $r_n \to \infty$. In Taubes's work, the asymptotic behavior of these solutions is related to the dynamics of $X$. We consider the rather unexplored case of sequences of solutions whose energy is not uniformly bounded as $n\to\infty$. Our first main result shows that when the energy grows more slowly than $r_n^{1/2}$, the limiting nodal set of the solutions converges to an invariant set of the vector field $X$. The main tool we use is a novel maximum principle for the solutions with the key property that it remains valid in the unbounded energy case. As a byproduct, in the usual case of sequences of solutions with bounded energy, we obtain a new, more straightforward proof of Taubes's result on the existence of periodic orbits that does not involve a local analysis or the vortex equations. Our second main result proves that, contrary to what happens in the bounded energy case, when the energy is unbounded there are no local restrictions to the limiting measures that may arise in the modified Seiberg-Witten equations. Furthermore, we obtain a connection about the dimension of the support of the limiting measure (as expressed through a $d$-Frostman property) and the energy growth of the sequence of local solutions we construct.

math.AP↗

arXiv2021

Chaos in the incompressible Euler equation on manifolds of high dimension

We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field $X$ on a closed manifold $N$, there is a Riemannian manifold $M$ on which the following holds: $N$ is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on $M$) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small perturbation of the vector field $X$ on $N$.

math.AP↗

arXiv2020

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

math.DG↗

arXiv2019

Periodic orbits of analytic Euler fields on 3-manifolds

On any closed Riemannian 3-manifold which is not a torus bundle, every nonvanishing analytic solution of the stationary Euler equations has a periodic trajectory. This result is originally due to A. Rechtman (arXiv:0904.2719) and K. Cieliebak and E. Volkov (arXiv:1402.6484); here we present an alternative proof of it.

math.DG↗

arXiv2018

High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology

Let $Σ$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $Σ$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $ψ$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $Δψ=-λψ$ (with $λ=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $ψ$ given by~$Σ$, up to an ambient diffeomorphism.

math.AP↗

arXiv2017

Geometric structures in the nodal sets of eigenfunctions of the Dirac operator

We show that, in round spheres of dimension $n\geq3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{Σ_1,...,Σ_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $ψ=(ψ_1,...,ψ_N)$ of the Dirac operator such that each submanifold $Σ_a$, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component $ψ_a$. The result holds for any choice of trivialization of the spinor bundle. The emergence of these structures takes place at small scales and sufficiently high energies.

math.DG↗

arXiv2016

Helicity is the only integral invariant of volume-preserving transformations

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mathcal I$ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.

math.DS↗

arXiv2015

Knotted structures in high-energy Beltrami fields on the torus and the sphere

Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset of T^3. In this paper we show that for any sufficiently large odd integer λthere exists a Beltrami field on S^3 or T^3 satisfying curl u = λu and with a collection of vortex lines and vortex tubes given by S, up to an ambient diffeomorphism.

math.AP↗