### Citations

227 |
Global Stability of Dynamical Systems
- Shub
- 1986
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Citation Context ... implies the existence of Lipshitz functions wuc : Duc Ñ D and wsc : Dsc Ñ D, with W scpF q “ tx “ wscpy, zqu, W ucpF q “ W scpinvpF qq “ ty “ wucpx, zqu. Then standard arguments (see Theorem 5.18 in =-=[Shu87]-=-) implies these functions are C1. The fact that µ ą 1 and TZW scpF q P Cscµ´1pZq, TZW upF q P Cscµ´1pZq implies W scpF q and W ucpF q intersect transversally, and W scpF q XW ucpF q is a graph over th... |

202 | Instability of dynamical systems with several degrees of freedom - Arnold - 1964 |

194 | Action minimizing invariant measures for positive definite Lagrangian systems. - Mather - 1991 |

184 |
Neishtadt.Mathematical Aspects of Classical and Celestial Mechanics
- Arnol’d, Kozlov, et al.
- 1988
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Citation Context ...s and expect to prove Arnold diffusion along this net in a future publication. The statement that Hsp0,Bst approximates H s p0,B is related to the classic result of partial averaging (see for example =-=[AKN06]-=-). The statement minkPΛzΛst |k| " maxkPΛst |k| says that the resonances in Λst is much stronger than the rest of the resonances in Λ. Partial averaging says that the weaker resonances contributes to s... |

144 | Small denominators and problems of stability of motion in classical and celestial mechanics - Arnold - 1963 |

97 |
Variational construction of connecting orbits
- Mather
- 1993
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Citation Context ... barrier is hLpϕ, ψq “ limTÑ8 hTLpϕ, ψq. The limit exists, and the function hL is Lipschitz in both variables. Denote hL,c “ hL´c¨v. Mather, Aubry and Mañe sets. These sets are defined by Mather (see =-=[Mat93]-=-). Here we only introduce the projected version. Define the Aubry and the Mañe sets as ALpcq “ tx P Td : hL,cpx, xq “ 0u, NLpcq “ " y P Td : min x,zPALpcq phL,cpx, yq ` hL,cpy, zq ´ hL,cpx, zqq “ 0 * ... |

72 | Lagrangian flows: the dynamics of globally minimizing orbits, International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé
- Mañé
- 1996
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Citation Context ... Rd is called a closed measure (see [Sor10], Remark 4.40) if for all f P C1pTdq,ż dfpϕq ¨ v dµpϕ, vq “ 0. This notion is equivalent to the more well known notion of holonomic measure defined by Mañe (=-=[Mañ97]-=-). For c P H1pTd,Rq » Rd, the alpha function αLpcq “ ´ inf ν ż pLpϕ, vq ´ c ¨ vqdνpϕ, vq, 27 where the minimization is over all closed Borel probability measures. When L “ LH we also use the notation ... |

64 | A geometric mechanism for diffusion in hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model - Delshams, Llave, et al. - 2003 |

48 | Existence of diffusion orbits in a priori unstable hamiltonian systems - Cheng, Yan - 2003 |

40 |
Variational construction of orbits of twist diffeomorphisms
- Mather
- 1991
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Citation Context ... is a transition chain such that ω0 “ a and ωN “ b for some N and showed that this implies existence of orbits asymptotic to T2b in the future and to T2a in the past. Generalized transition chains In =-=[Mat91a]-=- Mather proposed a diffusion mechanism where invariant tori where replaced by Aubry-Mather invariant sets for twist maps. For fiber convex superliear time-periodic Hamiltonians Hpθ, p, tq, θ, t P T, p... |

36 |
Instability of dynamical systems with many degrees of freedom
- Arnold
- 1964
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Citation Context ...here Id denotes the identity matrix. The main motivation behind this work is the question of Arnold diffusion, that is, topological instability for the system Hε. Arnold provided the first example in =-=[Arn64b]-=-, and asks ([Arn63; Arn64a; Arn94]) whether topological instability is “typical” in nearly integrable systems with n ě 2 (the system is stable when n “ 1, due to low dimensionality). It is well known ... |

36 | Evolution of slow variables in a priori unstable Hamiltonian systems - Treschev - 2003 |

31 | A functional analysis approach to Arnold diffusion - Berti, Bolle |

28 |
On minimizing measures of the action of autonomous Lagrangians,
- Carneiro
- 1995
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Citation Context ... Rn ˆ TÑ Rn the natural projection onto the action component. Recall that homology and cohomology are related by Legendre-Fenichel tranform LFβphq Ă H1pTn,Rq (see (5.1)). By a result of Diaz Carneiro =-=[Car95]-=- for each cohomology c P H1pTn,Rq the Aubry set Apcq Ă Sαpcq. Definition. Let ρ ą 0. We say that pH, h, ρq has the AM property if for any λ such that c P LFβpλhq and αHpcq ě ρ the Aubry set Apcq is a ... |

26 | Transition tori in the five-body problem - Moeckel - 1996 |

20 | Arnold diffusion in Hamiltonian systems: a priori unstable case - Cheng, Yan |

17 | A strong form of Arnold diffusion for two and a half degrees of freedom, preprint arXiv:1212.1150
- Kaloshin, Zhang
- 2012
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Citation Context ...ssing, while the grey dots requires switching of a particular slow system defined on Tn ˆ Rn, denoted Hsp0,B. More precisely, in an Op?εq´neighborhood of p0, the system Hε admits the normal form (see =-=[KZ13]-=-, Appendix B) Hsp0,Bpϕ, Iq ` ? εP pϕ, I, τq, ϕ P Tn, I P Tn, τ P ?εT, where ϕi “ ki ¨ pθ, tq, 1 ď i ď n, pp´ p0q{? “ k̄1I1 ` ¨ ¨ ¨ k̄nIn. Therefore H is conjugate to a fast periodic perturbation to ... |

16 |
Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders. Preprint available at http://arxiv.org/abs
- Bernard, Kaloshin, et al.
- 2011
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Citation Context ...ain 2-torus T2c , i.e. for some submersion pic : Tn ˆBn ˆ TÑ T2 we have that piApcq : Apcq Ñ T2c is one-to-one and the inverse is Lischitz. This is similar but more involved than what is presented in =-=[BKZ11]-=-. See discussion of n “ 3 in [KZ14]. • 3-dimensional cylinders rC 3i for H correspond to 2-dimensional cylinders C2 for averaged Hamiltonians. • The cylinder rC 3i might consists of several connected... |

15 | Growth of Sobolev norms in the cubic defocusing nonlinear Schrodinger equation. - Guardia, Kaloshin - 2012 |

15 |
diffusion I: Announcement of results
- Arnold
- 2002
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Citation Context ...stem is a 2 degrees of freedom mechanical system, the structure of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see =-=[Mat03]-=-, [Mat08], [Mat11], [Che13], [KZ13],[GK14b], [KMV04], [Mar12a], [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. ... |

14 | Young measures, superposition and transport.
- Bernard
- 2008
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Citation Context ...nd is presented in Section 6.2. In Section 6.1 we prove item 1, 3 and 4. 6.1 The Mañe set and barrier function We first state an alternate definition of the Aubry and Mañe sets due to Fathi (see also =-=[Ber08]-=-). Let u be a weak KAM solution for the Lagrangian L. We define GpL, uq to be the set of points pϕ, vq P Td ˆRd such that there exists a pu, L, αLq-calibrated curve γ : p´8, 0s Ñ Td, such that pϕ, vq ... |

14 | Lecture notes on Mather’s theory for Lagrangian systems.
- Sorrentino
- 2010
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Citation Context ...n b´ a such that } 9γ} ď D ([Fat08] Corollary 4.3.2). This property is called the a priori compactness. The alpha function and minimal measures. A measure µ on Td ˆ Rd is called a closed measure (see =-=[Sor10]-=-, Remark 4.40) if for all f P C1pTdq,ż dfpϕq ¨ v dµpϕ, vq “ 0. This notion is equivalent to the more well known notion of holonomic measure defined by Mañe ([Mañ97]). For c P H1pTd,Rq » Rd, the alpha ... |

13 | Mathematical problems in classical physics. Trends and perspectives in applied mathematics - Arnold - 1994 |

11 | An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension - Kaloshin, Saprykina |

9 | On the Conley decomposition of Mather sets.
- Bernard
- 2010
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Citation Context ...in Appendix A.1 6 The formulation of the limit theorem in weak KAM solution requires special care. A natural candidate is Tonelli convergence (convergence of Lagrangian within the Tonelli family, see =-=[Ber10]-=-). In our setup, Hsp0,Bst and H s p0,B are defined on different spaces, we need to consider the trivial lift of Hsp0,Bst to a higher dimensional space. The lifted Lagrangian is then degenerate and obv... |

9 |
The Weak KAM theorem in Lagrangian dynamics. 10th Preliminary version
- Fathi
- 2009
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Citation Context ...t ą 0, we have upxq “ inf yPTd,γp0q“y,γptq“x ˆ upyq ` ż t 0 pLHpγptq, 9γptqq ´ c ¨ 9γptq ` αHpcqqdt ˙ , where γ : r0, ts Ñ Td is absolutely continuous. Weak KAM solutions exist and are Lipschitz (see =-=[Fat08]-=-, [Ber10]). • The relation between Lagrangians We now turn to the weak KAM solutions of dominant Hamiltonians. Fix Bst and consider pBwk, p, U st,Uwkq P Ωm,dpBstq and write Hs “ HspBst,Bwk, p, U st,Uw... |

8 |
Shadowing property of geodesics in Hedlund’s metric, Ergodic Theory Dynam
- Levi
- 1997
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Citation Context ...tation vectors. Indeed, generically minimal sets with rational rotation vector is a hyperbolic periodic orbit. In the case d ą 2 it is not longer true as the well-know Hedlund example shows (see e.g. =-=[Lev97]-=-). More exactly, if we consider an integer homology h on Td and consider infinite minimizers of homology class h, i.e. the Aubry set Aphq (see section 5.1 for precise definition). Then • Aphq does not... |

8 |
Lectures on the Geometry of Numbers," Notes by B. Friedman, Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter, With a preface by Chandrasekharan
- Siegel
- 1989
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Citation Context ...inition of Mi`1, i.e |k1i`1| “Mi`1. We have |k1i| “Mi, m ă i ď d, |k1j| ď |k1i|, m ă j ă i ď d, but k11, ¨ ¨ ¨ , k1d may not form a basis. We turn them into a basis using the following procedure (see =-=[Sie89]-=-). For each j “ 1, ¨ ¨ ¨ ,m, define cj “ mintsj : sj,1k11 ` ¨ ¨ ¨ ` sj,j´1k1j´1 ` sjk1j P Λ, sj P R`, sj,i P R` Y t0uu. (3.1) We define cj,j´1 using a similar minimization given the value cj: cj,j´1 “... |

7 | Instability of high dimensional hamiltonian systems: Multiple resonances do not impede diffusion - Delshams, Llave, et al. - 2013 |

7 |
Generic hyperbolic properties of classical systems on the torus T 2
- Marco
- 2012
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Citation Context ...re of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11], [Che13], [KZ13],[GK14b], [KMV04], =-=[Mar12a]-=-, [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is proposed that we can sidestep this difficulty ... |

7 |
Shortest curves associated to a degenerate Jacobi metric on T 2
- Mather
- 2011
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Citation Context ...s of freedom mechanical system, the structure of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], =-=[Mat11]-=-, [Che13], [KZ13],[GK14b], [KMV04], [Mar12a], [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is pr... |

6 |
On the type of certain periodic orbits minimizing the Lagrangian action Nonlinearity 11
- Arnaud
- 1998
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Citation Context ... have homology h (see [Lev97]). • In the class of Tonelli Hamiltonians minimization within the class of closed loops in some homology class h might lead to non-hyperbolic minimal periodic orbits (see =-=[Arn98]-=-). The AM property guarantee all these properties. The jump. In [KZ13] section 12 we show that for each pair of “crossing” cylinders there is a jump from an invariant set from one generalized transiti... |

6 |
Generic hyperbolic properties of nearly integrable systems on A3
- Marco
- 2012
(Show Context)
Citation Context ...(minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11], [Che13], [KZ13],[GK14b], [KMV04], [Mar12a], =-=[Mar12b]-=-). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is proposed that we can sidestep this difficulty by using d... |

6 |
Examples of Aubry sets
- Mather
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Citation Context ..., i.e. the Aubry set Aphq (see section 5.1 for precise definition). Then • Aphq does not have to consist of periodic orbits or does not even have to have countably many invariant components (see e.g. =-=[Mat04]-=-). • Aphq consisting of periodic orbits do not guarantee they have homology h (see [Lev97]). • In the class of Tonelli Hamiltonians minimization within the class of closed loops in some homology class... |

6 |
Order structure on action minimizing orbits Symplectic Topology and Measure Preserving Dynamical Systems (Contemporary Mathematics vol 512)
- Mather
- 2010
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Citation Context ...homology g is trivial in the weak variable. 57 3. Let us explain the proof briefly. Property A1 is a known result. This property is used in Arnold diffusion in 212 degrees of freedom, and we refer to =-=[Mat10]-=-, [Mat11], [KZ13], [Che13] for more details. 4. Property A2 uses the first type of dimension reduction. The assumption ensures that Hst admits at most two minimal hyperbolic saddles. An arbitrarily sm... |

6 |
The stable manifold theorem via an isolating block. Symposium on Ordinary Differential Equations (Univ
- McGehee
- 1972
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Citation Context ...olic invariant manifolds via isolation block We state an abstract theorem on existence of normally hyperbolic invariant manifolds for a smooth map F . based on Conley’s isolation blocks (see McGehee, =-=[McG73]-=-). We introduce a set of notations. We have three components x P Rs, y P Ru, z P Ωc Ă Rc , where Ωc is a (possibly unbounded) convex set. We assume that Ωc admits a C1 complete Riemannian metric g. We... |

6 | diffusion far from strong resonances in multidimensional a priori unstable hamiltonian systems - Arnold |

5 |
Instability of totally elliptic points of symplectic maps in dimension 4
- Kaloshin, Mather, et al.
(Show Context)
Citation Context ...e structure of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11], [Che13], [KZ13],[GK14b], =-=[KMV04]-=-, [Mar12a], [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is proposed that we can sidestep this d... |

4 | Orbits of nearly integrable systems accumulating to KAM tori. Preprint available at http://arxiv
- Guardia, V
- 2014
(Show Context)
Citation Context ...ystem, the structure of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11], [Che13], [KZ13],=-=[GK14b]-=-, [KMV04], [Mar12a], [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is proposed that we can sidest... |

3 |
The gradient structure of a flow. I”. In: Ergodic Theory Dynam. Systems 8˚.Charles Conley Memorial Issue
- Conley
- 1988
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Citation Context ...¨ ¨ , TN´1 ě T , such that dpφTixi, xi`1q ă ε. We say that xCXy if for any ε, T ą 0, there exists an pε, T q´chain with x0 “ x and xN “ y. The relation CX is called the chain transitive relation (see =-=[Con88]-=-). The family of maps φ̄t “ φt defines a semi-flow on the set GpL´ c ¨ v, uq, and therefore defines a chain transitive relation. Given ϕ, ψ P Td and a weak KAM solution u of L ´ c ¨ v, we say that ϕCu... |

3 | diffusion for a priori unstable systems and a five-body problem - Arnold - 2010 |

2 |
diffusion in nearly integrable systems”. In: preprint
- “Arnold
- 2013
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Citation Context ...dom mechanical system, the structure of its (minimal) orbits is well understood. This fact underlies the results on Arnold diffusion in two and half degrees of freedom (see [Mat03], [Mat08], [Mat11], =-=[Che13]-=-, [KZ13],[GK14b], [KMV04], [Mar12a], [Mar12b]). This is no longer the case when n ą 2, which is a serious obstacle to proving Arnold diffusion in higher degrees of freedom. In [KZ14] it is proposed th... |

2 |
diffusion for three and half degrees of freedom”. In: preprint http://www2.math.umd.edu/ vkaloshi/papers/announcethree-and-half.pdf
- “Arnold
- 2014
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Citation Context ... the only important missing part of construction of diffusing orbits along the path Γ is the jump. Construction of a variational problem leading to the jump for 312 -degree of freedom is in section 8 =-=[KZ14]-=-. C Normally hyperbolic invariant manifolds In this section we state a version of the center manifold theorem and prove Corollary 2.3. While the central manifold theorem is classical, we need an versi... |

2 |
Hyperbolic sets near homoclinic loops to a saddle for systems with a first integral”. In: Regular and Chaotic Dynamics to appear
- Turaev
- 2014
(Show Context)
Citation Context ... in general and normally hyperbolic invariant manifolds (NHIM) for critical energy for simple homologies is discussed in [KZ14], sect. 6.3. We expect these methods extend to arbitrary n ě 3 (see also =-=[Tur14]-=-). • We point out that presence of NHIC and NHIM is still not sufficient for diffusion as we need to construct the jump from one homology to another (see sect. 12 [KZ13]). In the case n “ 3 it require... |

1 |
Introduction to Dynamical systems”. In: Introduction to Dynamical systems
- Brin, Stuck
- 2002
(Show Context)
Citation Context ...and both Λ and BΛ are invariant under φt0 . A more common definition of normally hyperbolic (fully) invariant cylinders assumes a spectral radius condition, but our definition is equivalent, see e.g. =-=[BS02]-=- Prop.5.2.2. Moreover: • If the parameters α, β satisfies the bunching condition α ă β2, then the bundles Es, Eu are C1 smooth. • When Es, Eu are smooth, we can always choose the adapted metric g such... |