Lagrangian flows: The dynamics of globally minimizing orbits-II

Contreras, Gonzalo; Delgado, Jorge; Iturriaga, Renato

doi:10.1007/bf01233390

Cited by 74 publications

(99 citation statements)

References 7 publications

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“…The problem we consider here is in some sense analogous (although we do not consider the homological position) to the problems considered in Aubry-Mather theory (see [6,18]) for Lagrangian flows. A recent result of Mañé (see [18] and also [5,6]) on Lagrangian flows shows that generically on the Lagrangian there is a unique measure minimizing…”

Section: A Measure In M(f ) Is Called a Lyapunov Minimizing Measurementioning

confidence: 99%

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Lyapunov minimizing measures for expanding maps of the circle

Contreras

Lopes

Thieullen³

2001

Ergod. Th. Dynam. Sys.

141

247

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Abstract. We consider the set of maps f ∈ F α+ = ∪ β>α C 1+β of the circle which are covering maps of degree D, expanding, min x∈S 1 f (x) > 1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f -invariant probability measure µ minimizing ln f dµ over all f -invariant probability measures. We show there exists a set G + ⊂ F α+ , open and dense in the C 1+α -topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C 1+α -topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8].We will also present some results on the set of f -invariant measures µ maximizing A dµ for a fixed C 1 -expanding map f and a general potential A, not necessarily equal to −ln f .

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Section: A Measure In M(f ) Is Called a Lyapunov Minimizing Measurementioning

confidence: 99%

“…A recent result of Mañé (see [18] and also [5,6]) on Lagrangian flows shows that generically on the Lagrangian there is a unique measure minimizing…”

Section: A Measure In M(f ) Is Called a Lyapunov Minimizing Measurementioning

confidence: 99%

Lyapunov minimizing measures for expanding maps of the circle

Contreras

Lopes

Thieullen³

2001

Ergod. Th. Dynam. Sys.

141

247

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“…Let π : T M → M the canonical projection. Then π| b Σ : Σ → M is a bijective map with Lipschitz inverse, the proof of this property can be seen in [8,theorem VI] and is a extension of the Mather's graph theorem [25, theorem 2].…”

Section: 3mentioning

confidence: 99%

“…We recall the definition of the strict Mañé's critical value for convex and superlinear lagrangians (cf. [23,8] It is well known that for an arbitrary surface M , if Ω = dη and c > c 0 (L η ), then the restriction of the exact magnetic flow in the energy set T c M is a reparametrization of a geodesic flow in the unit tangent bundle for an appropriate Finsler metric on M (cf. [9]).…”

Section: Introduction and Statementsmentioning

confidence: 99%

Positive topological entropy for magnetic flows on surfaces

Miranda¹

2007

Nonlinearity

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Abstract. We study the topological entropy of the magnetic flow on a closed riemannian surface. We prove that if the magnetic flow has a non-hyperbolic closed orbit in some energy set T c M = E −1 (c), then there exists an exact C ∞ -perturbation of the 2-form Ω such that the new magnetic flow has positive topological entropy in T c M . We also prove that if the magnetic flow has an infinite number of closed orbits in T c M , then there exists an exact C 1 -perturbation of Ω with positive topological entropy in T c M . The proof of the last result is based on an analog of Franks' lemma for magnetic flows on surfaces, that is proven in this work, and Mañé's techniques on dominated splitting. As a consequence of those results, an exact magnetic flow on S 2 in high energy levels admits a C 1 -perturbation with positive topological entropy. In the appendices we show that an exact magnetic flow on the torus in high energy levels admits a C ∞ -perturbation with positive topological entropy.

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“…In fact, it follows ( [22], [10]) that D E (x, y) = −∞ for E < E and D E (x, x) = 0 for E ≥ E and any x, y ∈ M .…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for optimal mass transportation

Wolansky

2011

Calc. Var.

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The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures λ + , λ − of equal masswhere W p , p ≥ 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.

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Lagrangian flows: The dynamics of globally minimizing orbits-II

Cited by 74 publications

References 7 publications

Lyapunov minimizing measures for expanding maps of the circle

Lyapunov minimizing measures for expanding maps of the circle

Positive topological entropy for magnetic flows on surfaces

Limit theorems for optimal mass transportation

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