[논문 리뷰] Generic Ma\~n\'e sets

Gonzalo Contreras|arXiv (Cornell University)|2014. 10. 27.

Mathematical Dynamics and Fractals참고 문헌 31인용 수 4

한 줄 요약

이 논문은 컴acts한 다양체 위의 토넬리 라그랑지안에 대해 $C^2$ 일반적인 초구형 마네 집합이 순환 궤도를 포함함을 증명한다. 차원 2인 경우, 이전의 일반적인 초구형성 결과와 결합하면 마네의 추측을 확인할 수 있다: $C^2$ 일반적인 포텐셜 $f$에 대해 $L + f$의 마네 집합은 초구형 순환 궤도이며, 이는 최소 동역학이 초구형 순환 궤도에 지지되는 밀도 있는 라그랑지안의 집합을 확립한다.

ABSTRACT

Abstract. We prove that C2 generic hyperbolic Mañe ́ sets contain a periodic orbit. In dimension 2, adding a result with A. Figalli and L. Rifford [9], we obtain Mañé’s Conjecture [18] for surfaces in the C2 topology. Let M be a closed riemannian manifold. A Tonelli Lagrangian is a C2 function L: TM → R such that it is (i) Convex: ∀(x, v) ∈ TM, ∂2vvL(x, v) is positive definite. (ii) Superlinear: ∀A> 0 ∃B> 0 such that ∀(x, v) ∈ TM: L(x, v)> A |v|x −B. Given k ∈ R, the Mañe ́ potential is defined as φk: M ×M → R ∪ {−∞} as φk(x, y): = inf γ∈C(x,y) k + L(γ, γ̇), where C(x, y): = {γ: [0, T]→M absolutely continuous | T> 0, γ(0) = x, γ(T) = y}. The Mañe ́ critical value is c(L): = sup { k ∈ R | ∃x, y ∈M: φk(x, y) = − ∞}. See [11] for several characterizations of c(L). A curve γ: R→M is semistatic if ∀s < t: ∫ t s c(L) + L(γ, γ̇) = φc(L)(γ(s), γ(t)). The Mañe ́ set of L is N ̃ (L): = {(γ(t), γ̇(t)) ∈ TM | t ∈ R, γ: R→M is semistatic}. The Euler-Lagrange equation d

연구 동기 및 목표

• To establish that $C^2$ generic hyperbolic Ma\'n\'e sets contain periodic orbits.
• To prove that for compact surfaces, $C^2$ generic perturbations of Tonelli Lagrangians yield Ma\'n\'e sets that are hyperbolic periodic orbits.
• To confirm Ma\'n\'e's Conjecture in the $C^2$ topology for surfaces by showing the set of potentials for which the Ma\'n\'e set is a hyperbolic periodic orbit is open and dense.
• To demonstrate that hyperbolicity of the Ma\'n\'e set implies the existence of a subshift of finite type model for the dynamics near the set.
• To show that minimizing measures supported in the Ma\'n\'e set lift to invariant measures on a suspension flow over a subshift, linking dynamics to symbolic dynamics.

제안 방법

• Uses upper semicontinuity of the Ma\'n\'e set map $\phi \mapsto \widetilde{\mathcal{N}}(L + \phi)$ in the $C^2$ topology to relate perturbations to structural stability.
• Applies results from symbolic dynamics and shadowing theory to approximate hyperbolic sets by periodic orbits.
• Employs a radial projection to identify energy levels $E^{-1}\{c(L)\}$ with the unit tangent bundle $SM$ for small perturbations.
• Constructs a continuous family of $C^1$ flows on $SM$ via conjugation with radial projections, enabling the use of hyperbolicity theory.
• Uses the stability of hyperbolic sets under small $C^2$ perturbations to lift the dynamics of $\widetilde{\mathcal{N}}(L)$ to a suspension flow over a subshift of finite type.
• Applies structural stability and expansivity arguments to show that hyperbolic periodic orbits persist under small perturbations of the potential.

실험 결과

연구 질문

• RQ1 Does the $C^2$ generic hyperbolic Ma\'n\'e set for a Tonelli Lagrangian contain a periodic orbit?
• RQ2 Is the set of $C^2$ potentials for which the Ma\'n\'e set of $L + \phi$ is a hyperbolic periodic orbit open and dense on compact surfaces?
• RQ3 Can the dynamics of a hyperbolic Ma\'n\'e set without fixed points be modeled by a suspension flow over a subshift of finite type?
• RQ4 How does the Ma\'n\'e set behave under small $C^2$ perturbations of the potential $\phi$?
• RQ5 What is the relationship between minimizing measures supported in the Ma\'n\'e set and invariant measures on a symbolic system?

주요 결과

• The set $\mathcal{H}^2(L_0)$ of $C^2$ potentials for which $\widetilde{\mathcal{N}}(L_0 + \phi)$ is hyperbolic is open, and its closure contains all potentials for which the Ma\'n\'e set contains a periodic orbit.
• For $\dim M = 2$, the set $\mathcal{H}^2(L_0)$ is open and dense in $C^2(M, \mathbb{R})$, confirming Ma\'n\'e's Conjecture in this setting.
• The set $\mathcal{HP}^2(L_0)$ of potentials for which $\widetilde{\mathcal{N}}(L_0 + \phi)$ is a hyperbolic periodic orbit or singularity is open and dense in $\mathcal{H}^2(L_0)$.
• Hyperbolic Ma\'n\'e sets without fixed points admit a topological model as a suspension flow over a subshift of finite type, with continuous conjugacy to the flow restricted to the energy level.
• Any invariant measure supported in a neighborhood of the Ma\'n\'e set for $L + \phi$ lifts to an invariant measure on the suspension flow, and is minimizing if and only if its lift minimizes the action of $L + \phi$.
• The energy level $E_{L}^{-1}\{c(L)\}$ is diffeomorphic to $SM$ when $\widetilde{\mathcal{N}}(L)$ contains no fixed points, enabling a uniform treatment of the dynamics via the unit tangent bundle.