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Jinxin Xue

Tsinghua University, Mathematics, Faculty Member

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I am Xiaomi endowed professor at Tsinghua University. I work on dynamics related problems coming from various fields such as celestial mechanics, general relativity, symplectic geometry, geometric flow, etc.
Supervisors: Dmitry Dolgopyat

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Natalie Baddour

University of Ottawa | Université d'Ottawa

University of California, Merced

Artur Sergyeyev

Silesian University in Opava

Kansas State University

Case Western Reserve University

Dublin Institute of Technology

Okinawa Institute of Science and Technology

Mikail F Lumentut

National Taiwan University

Tilburg University

Hadia Hassan Selim

National Research Institute Of Astronomy And Geophysics Nriag

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Papers by Jinxin Xue

Some recent developments in Arnold diffusion

Cambridge University Press eBooks, May 9, 2024

KAM and geodesic dynamics of blackholes

arXiv (Cornell University), Dec 5, 2020

In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of boun... more In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena: the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations. We discover there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

A C^1 Arnol'd-Liouville theorem

Astérisque, 2020

EXISTENCE OF NONCONTRACTIBLE PERIODIC ORBITS OF HAMILTONIAN SYSTEM SEPARATING TWO LAGRANGIAN TORI ON T T n WITH APPLICATION TO NON CONVEX HAMILTONIAN SYSTEMS

arXiv (Cornell University), Aug 22, 2014

In this paper, we show the existence of non-contractible periodic orbits in Hamiltonian systems d... more In this paper, we show the existence of non-contractible periodic orbits in Hamiltonian systems defined on T * T n separating two Lagrangian tori under a certain cone assumption. Our result gives a positive answer to a question of Polterovich in [P]. As an application, we find periodic orbits in almost all the homotopy classes on a dense set of energy levels in Lorentzian type mechanical Hamiltonian systems defined on T * T 2. This solves a problem of Arnold in [A].

Closing lemma and KAM normal form

arXiv (Cornell University), Jul 13, 2022

In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. T... more In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing C 1 perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's C 1closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting. We also discuss the relation of our models to the recent researches on many-body localization in physics.

Super-hyperbolic orbits and non-collision singularities in a four-body problem

arXiv (Cornell University), Feb 23, 2023

Dynamics of composite symplectic Dehn twists

arXiv (Cornell University), Aug 27, 2022

This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensio... more This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensional topology, etc. We show that composite symplectic Dehn twists have certain form of nonuniform hyperbolicity that can be used to give positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, as signatures of pseudo Anosov mapping classes. Moreover, we also show that the rank of the Floer cohomology group grows exponentially under iterations.

On the Painlevé conjecture

Current Developments in Mathematics

Generic mean curvature flows with cylindrical singularities

Cornell University - arXiv, Oct 2, 2022

In this paper, we study the dynamics of the mean curvature flow approaching a cylindrical singula... more In this paper, we study the dynamics of the mean curvature flow approaching a cylindrical singularity and show that generically such a singularity is nondegenerate, robust, and isolated. This gives also consequences such as generic low regularity of level set flows, generic rotational MCFs being type-I, etc. Another discovery of the paper is the formulation of the "firecracker" type singularity and its role playing in the study of generic isolatedness of cylindrical singularities. The work appears as the confluence of methods and ideas from geometric flows, dynamical systems, and semilinear heat equations.

Generic regularity of Level Set Flows with spherical singularity

Sphere is known to be the only generic compact shrinker for mean curvature flow (MCF). In this pa... more Sphere is known to be the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, this gives that generically the arrival time function of level set flow with spherical singularity has at most C 2 regularity.

A New Mechanism for Noncollision Singularities

by Joseph Gerver and Jinxin Xue

In this paper, we prove the existence of noncollision singularities in the planar four-body probl... more In this paper, we prove the existence of noncollision singularities in the planar four-body problem with a model different from [19]. In this model, the acceleration can be arbitrarily fast and the masses can be comparable. This work provides a general principle to construct noncollision singularities as well as other related orbits with complicited dynamics. It not only answers a question in [19] but also solves an analogous version of a conjecture of Anosov.

Initial Perturbation of the Mean Curvature Flow for closed limit shrinker

This is a contribution to the program of dynamical approach to mean curvature flow initiated by C... more This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we pursue the stream of ideas of <cit.> and get a slight refinement of their results. We apply the invariant manifold theory from hyperbolic dynamics to study the dynamics close to a closed shrinker that is not a sphere. In the second theorem, we show that if a hypersurface under the rescaled mean curvature flow converges to a closed shrinker that is not a sphere, then a generic perturbation on initial data would make the flow leave a small neighborhood of the shrinker and never come back.

Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on T^*^n with application to non convex Hamiltonian systems

In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems d... more In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on T^*^n separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in P in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on T^*^2. This solves a problem of Arnold in A.

A C1 Arnol'd-Liouville theorem

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians... more In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom

by Cheng Chong-Qing and Jinxin Xue

In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex H... more In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: H(x,y)=h(y)+ P(x,y), x∈T^n, y∈R^n, n≥ 3. Under typical perturbation P, the system admits "connecting" orbit that passes through any finitely many prescribed small balls in the same energy level H^-1(E) provided E> h.

Rigidity of some abelian-by-cyclic solvable group actions on T^N

In this paper, we study a natural class of groups that act as affine transformations of T^N. We i... more In this paper, we study a natural class of groups that act as affine transformations of T^N. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on T^N while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by C^r diffeomorphims can be conjugated to the affine action by C^r-ϵ conjugacy. Next, we show that in any dimension, C^1 small perturbations can be conjugated to an affine action via C^1+ϵ conjugacy. The method is a generalization of the Herman th...

Arnold Diffusion in a Restricted Planar Four-Body Problem

This paper constructs a certain planar four-body problem which exhibits fast energy growth. The s... more This paper constructs a certain planar four-body problem which exhibits fast energy growth. The system considered is a quasi-periodic perturbation of the Restricted Planar Circular three-body Problem (RPC3BP). Gelfreich-Turaev's and de la Llave's mechanism is employed to obtain the fast energy growth. The diffusion is created by a heteroclinic cycle formed by two Lyapunov periodic orbits surrounding L_1 and L_2 Lagrangian points and their heteroclinic intersections. Our model is the first known example in celestial mechanics of the a priori chaotic case of Arnold diffusion.

A C^1 Arnol'd-Liouville theorem

arXiv: Dynamical Systems, 2016

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians... more In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Noncollision Singularities in a Planar Four-Body Problem

Abstract. In this paper, we show that there is a Cantor set of initial con-ditions in a planar fo... more Abstract. In this paper, we show that there is a Cantor set of initial con-ditions in a planar four-body problem such that all the four bodies escape to infinity in finite time avoiding collisions. This proves the Painleve ́ conjecture for the four-body case, thus settles the conjecture completely.

Non-Collision singularities in the Planar two-Center-two-Body problem

In this paper, we study a model of simplified four-body problem called planar two-center-two-body... more In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers Q_1=(-χ,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses μ≪ 1. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlevé conjecture.

Some recent developments in Arnold diffusion

Cambridge University Press eBooks, May 9, 2024

KAM and geodesic dynamics of blackholes

arXiv (Cornell University), Dec 5, 2020

In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of boun... more In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena: the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations. We discover there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

A C^1 Arnol'd-Liouville theorem

Astérisque, 2020

EXISTENCE OF NONCONTRACTIBLE PERIODIC ORBITS OF HAMILTONIAN SYSTEM SEPARATING TWO LAGRANGIAN TORI ON T T n WITH APPLICATION TO NON CONVEX HAMILTONIAN SYSTEMS

arXiv (Cornell University), Aug 22, 2014

In this paper, we show the existence of non-contractible periodic orbits in Hamiltonian systems d... more In this paper, we show the existence of non-contractible periodic orbits in Hamiltonian systems defined on T * T n separating two Lagrangian tori under a certain cone assumption. Our result gives a positive answer to a question of Polterovich in [P]. As an application, we find periodic orbits in almost all the homotopy classes on a dense set of energy levels in Lorentzian type mechanical Hamiltonian systems defined on T * T 2. This solves a problem of Arnold in [A].

Closing lemma and KAM normal form

arXiv (Cornell University), Jul 13, 2022

In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. T... more In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing C 1 perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's C 1closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting. We also discuss the relation of our models to the recent researches on many-body localization in physics.

Super-hyperbolic orbits and non-collision singularities in a four-body problem

arXiv (Cornell University), Feb 23, 2023

Dynamics of composite symplectic Dehn twists

arXiv (Cornell University), Aug 27, 2022

This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensio... more This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensional topology, etc. We show that composite symplectic Dehn twists have certain form of nonuniform hyperbolicity that can be used to give positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, as signatures of pseudo Anosov mapping classes. Moreover, we also show that the rank of the Floer cohomology group grows exponentially under iterations.

On the Painlevé conjecture

Current Developments in Mathematics

Generic mean curvature flows with cylindrical singularities

Cornell University - arXiv, Oct 2, 2022

In this paper, we study the dynamics of the mean curvature flow approaching a cylindrical singula... more In this paper, we study the dynamics of the mean curvature flow approaching a cylindrical singularity and show that generically such a singularity is nondegenerate, robust, and isolated. This gives also consequences such as generic low regularity of level set flows, generic rotational MCFs being type-I, etc. Another discovery of the paper is the formulation of the "firecracker" type singularity and its role playing in the study of generic isolatedness of cylindrical singularities. The work appears as the confluence of methods and ideas from geometric flows, dynamical systems, and semilinear heat equations.

Generic regularity of Level Set Flows with spherical singularity

Sphere is known to be the only generic compact shrinker for mean curvature flow (MCF). In this pa... more Sphere is known to be the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, this gives that generically the arrival time function of level set flow with spherical singularity has at most C 2 regularity.

A New Mechanism for Noncollision Singularities

by Joseph Gerver and Jinxin Xue

In this paper, we prove the existence of noncollision singularities in the planar four-body probl... more In this paper, we prove the existence of noncollision singularities in the planar four-body problem with a model different from [19]. In this model, the acceleration can be arbitrarily fast and the masses can be comparable. This work provides a general principle to construct noncollision singularities as well as other related orbits with complicited dynamics. It not only answers a question in [19] but also solves an analogous version of a conjecture of Anosov.

Initial Perturbation of the Mean Curvature Flow for closed limit shrinker

This is a contribution to the program of dynamical approach to mean curvature flow initiated by C... more This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we pursue the stream of ideas of <cit.> and get a slight refinement of their results. We apply the invariant manifold theory from hyperbolic dynamics to study the dynamics close to a closed shrinker that is not a sphere. In the second theorem, we show that if a hypersurface under the rescaled mean curvature flow converges to a closed shrinker that is not a sphere, then a generic perturbation on initial data would make the flow leave a small neighborhood of the shrinker and never come back.

Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on T^*^n with application to non convex Hamiltonian systems

In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems d... more In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on T^*^n separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in P in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on T^*^2. This solves a problem of Arnold in A.

A C1 Arnol'd-Liouville theorem

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians... more In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom

by Cheng Chong-Qing and Jinxin Xue

In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex H... more In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: H(x,y)=h(y)+ P(x,y), x∈T^n, y∈R^n, n≥ 3. Under typical perturbation P, the system admits "connecting" orbit that passes through any finitely many prescribed small balls in the same energy level H^-1(E) provided E> h.

Rigidity of some abelian-by-cyclic solvable group actions on T^N

In this paper, we study a natural class of groups that act as affine transformations of T^N. We i... more In this paper, we study a natural class of groups that act as affine transformations of T^N. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on T^N while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by C^r diffeomorphims can be conjugated to the affine action by C^r-ϵ conjugacy. Next, we show that in any dimension, C^1 small perturbations can be conjugated to an affine action via C^1+ϵ conjugacy. The method is a generalization of the Herman th...

Arnold Diffusion in a Restricted Planar Four-Body Problem

This paper constructs a certain planar four-body problem which exhibits fast energy growth. The s... more This paper constructs a certain planar four-body problem which exhibits fast energy growth. The system considered is a quasi-periodic perturbation of the Restricted Planar Circular three-body Problem (RPC3BP). Gelfreich-Turaev's and de la Llave's mechanism is employed to obtain the fast energy growth. The diffusion is created by a heteroclinic cycle formed by two Lyapunov periodic orbits surrounding L_1 and L_2 Lagrangian points and their heteroclinic intersections. Our model is the first known example in celestial mechanics of the a priori chaotic case of Arnold diffusion.

A C^1 Arnol'd-Liouville theorem

arXiv: Dynamical Systems, 2016

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians... more In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Noncollision Singularities in a Planar Four-Body Problem

Abstract. In this paper, we show that there is a Cantor set of initial con-ditions in a planar fo... more Abstract. In this paper, we show that there is a Cantor set of initial con-ditions in a planar four-body problem such that all the four bodies escape to infinity in finite time avoiding collisions. This proves the Painleve ́ conjecture for the four-body case, thus settles the conjecture completely.

Non-Collision singularities in the Planar two-Center-two-Body problem

In this paper, we study a model of simplified four-body problem called planar two-center-two-body... more In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers Q_1=(-χ,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses μ≪ 1. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlevé conjecture.