Chain Recurrence, Chain Transitivity, Lyapunov Functions and Rigidity of Lagrangian Submanifolds of Optical Hypersurfaces

Abbondandolo, Alberto; Bernardi, Olga; Cardin, Franco

doi:10.1007/s10884-016-9543-5

Cited by 6 publications

(16 citation statements)

References 20 publications

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“…An invariant function is also called a first integral of the system. There are several works that study the existence of non trivial (non constant) first integrals, see for instance [ABC16,FP15,FS04,Hur86,Man73,Pag11]. In this work we study dynamical conditions that imply the non-existence of first integrals.…”

Section: The Study Of Invariant Functions and Trivial Centralizersmentioning

confidence: 99%

On the centralizer of vector fields: criteria of triviality and genericity results

Obata

Santiago

2020

Math. Z.

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In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold M d has a "small" centralizer. In the C 1 case, we give two criteria, one of which is C 1 -generic, which guarantees that the centralizer of a C 1 -generic vector field is indeed small, namely collinear. The other criterion states that a C 1 separating flow has a collinear C 1 -centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as quasi-triviality. In particular, the C 1 -centralizer of a C 1 -generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a C 1 -generic vector field, which includes C 1 -generic Axiom A (or sectional Axiom A) vector fields and C 1 -generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is trivial (as small as it can be), and we show that in higher regularity, collinearity and triviality of the C d -centralizer are equivalent properties for a generic vector field in the C d topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity C 2 , the C 1 -centralizer is trivial. 9 4. Quasi-triviality 13 4.1. Collinear does not imply quasi-trivial 13 4.2. The case of hyperbolic zeros 13 5. The study of invariant functions and trivial centralizers 18 5.1. First integrals and trivial C 1 -centralizers 19 5.2. Some results in higher regularity 20 6. The generic case 23 Appendix A. The separating property is not generic 40 References

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Section: The Study Of Invariant Functions and Trivial Centralizersmentioning

confidence: 99%

On the centralizer of vector fields: criteria of triviality and genericity results

Obata

Santiago

2020

Math. Z.

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“…In particular, f is a first integral if and only if N (f ) = X. We refer to Lemma 1.5 in [1] for a characterization of Lyapunov functions and first integrals in the case of a flow induced by a locally Lipschitz continuous vector field.…”

Section: Definition (Lyapunov Function Neutral Set and First Integral)mentioning

confidence: 99%

“…Indeed, thanks to the arbitrariness of the parameter ε > 0 involved in both the definitions of recurrence, Fathi and Pageault equivalently described recurrent points as minima of appropriate functionals defined on the space of finite sequences of points. In particular, bearing in mind formulae (1) and (2), for strong chain recurrent points the functional is the sum of the amplitudes of the jumps; for chain recurrent points, the functional is the maximum of the amplitudes of the jumps. For a discrete dynamical system, if u is a Lyapunov function for g, that is u • g ≤ u, the neutral set of u is…”

Section: Definition (Lyapunov Function Neutral Set and First Integral)mentioning

confidence: 99%

“…Then (i) There exists a Lipschitz continuous Lyapunov function u for g such that N (u) = SCR(g). [1] it is not constructed a single Lyapunov function whose neutral set coincides with the strong chain recurrent set. We finally underline that, more recently, The proof of this theorem combines together variational and dynamical methods.…”

Section: Definition (Lyapunov Function Neutral Set and First Integral)mentioning

confidence: 99%

“…This proves that u is strict outside SCR(φ), that is N (u) ⊆ SCR(φ). However, by Proposition 1.6 in [1], the neutral set of every Lipschitz continuous Lyapunov function for φ contains the strong chain recurrent set of φ, so SCR(φ) ⊆ N (u). We then conclude that N (u) = SCR(φ).…”

Section: Let Us Introducesmentioning

confidence: 99%

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Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set

Bernardi

Florio

2018

Dynamical Systems

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The aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict -that is strictly decreasing-outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strictly decreasing outside the chain recurrent set of a continuous flow. We then present two consequences of this theorem. From one hand, we characterize the strong chain recurrent set in terms of Lipschitz continuous Lyapunov functions. From the other hand, in the case of a flow induced by a vector field, we establish a sufficient condition for the existence of a C 1,1 strict Lyapunov function and we also discuss various examples. Moreover, for general continuos flows, we show that the strong chain recurrent set has only one strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions are the constants. Finally, we provide a necessary and sufficient condition to guarantee that the strong chain recurrent set and the chain recurrent one coincide.

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Aubry–Mather Theory for Conformally Symplectic Systems

Marò

Sorrentino

2017

Commun. Math. Phys.

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In this article we develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.

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Chain Recurrence, Chain Transitivity, Lyapunov Functions and Rigidity of Lagrangian Submanifolds of Optical Hypersurfaces

Cited by 6 publications

References 20 publications

On the centralizer of vector fields: criteria of triviality and genericity results

On the centralizer of vector fields: criteria of triviality and genericity results

Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set

Aubry–Mather Theory for Conformally Symplectic Systems

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