### Abstract

We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝ^{d-1}, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

Original language | English |
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Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Mathematical Physics Electronic Journal |

Volume | 6 |

Publication status | Published - 1 Dec 2000 |

Externally published | Yes |

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**Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits.** / Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits

AU - Valdinoci, Enrico

PY - 2000/12/1

Y1 - 2000/12/1

N2 - We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

AB - We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

UR - http://www.scopus.com/inward/record.url?scp=0007017087&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 1

EP - 30

JO - Mathematical Physics Electronic Journal

JF - Mathematical Physics Electronic Journal

SN - 1086-6655

ER -