Rodrigo Pereira Pacheco. Destruction of Invariant Graphs by Cˆ{1,\beta} Perturbations. Tese de Doutorado

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1 Rodrigo Pereira Pacheco Destruction of Invariant Graphs by Cˆ{1,\beta} Perturbations Tese de Doutorado Thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática, PUC Rio, as partial fulfillment of the requirements for the degree of Doutor em Matemática. Advisor: Prof. Rafael Oswaldo Ruggiero Rodriguez Rio de Janeiro March 2014

2 Rodrigo Pereira Pacheco Destruction of Invariant Graphs by Cˆ{1,\beta} Perturbations Thesis presented to the Programa de Pós-Graduação em Matemática of the Departamento de Matemática do Centro Técnico Científico da PUC Rio, as partial fulfillment of the requirements for the degree of Doutor. Prof. Rafael Oswaldo Ruggiero Rodriguez Advisor Departamento de Matemática - PUC-Rio Prof. Mário Jorge Dias Carneiro Departamento de Matemática - UFMG Prof. Umberto Leone Hryniewicz Departamento de Matemática - UFRJ Prof. Clodoaldo Grotta Ragazzo Instituto de Matemática - USP Prof. José Antônio Gonçalves Miranda Departamento de Matemática - UFMG Prof. José Barbosa Gomes Instituto de Matemática - UFJF Prof. David Francisco Martínez Torres Departamento de Matemática - PUC-Rio Prof. José Eugenio Leal Coordinator of the Centro Técnico Científico - PUC-Rio Rio de Janeiro, March 28 th,2014

3 All rights reserved. Rodrigo Pereira Pacheco B.A. in Mathematics, from Universidade do Espírito Santo and M.A. in Mathematics, from Pontfícia Universidade Católica do Rio de Janeiro. Pacheco, Rodrigo P. Bibliographic data Destruction of Invariant Graphs by Cˆ{1,\beta} Perturbations / Rodrigo Pereira Pacheco; advisor: Rafael Oswaldo Ruggiero Rodriguez f. : il. ; 30 cm Tese (Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro, Departamento de Matemática, Inclui bibliografia 1. Matemática Teses. 2. Geometria Finsler conforme.; 3. Equação de Jacobi. ; 4. Gráficos Lagrageanos. ; 5. Equação de Euler-Lagrange. ; 6. Pontos conjugados.. I. Pacheco, Rodrigo P.. II. Pontifícia Universidade Católica do Rio de Janeiro. Departamento de Matemática. III. Título. CDD: 510

4 To Sabrina and Maria Stella.

5 Acknowledgments First and foremost I would like to thank God. The faith I have in you was the light that guided me through these years. I would like to express my sincere gratitude to my advisor Rafael O. Ruggiero for his availability, patience and for teaching me a lot of interesting things. I would like to thank my committee members, professor Mário Jorge D. Carneiro, professor Umberto L. Hryniewicz, professor Clodoaldo G. Ragazzo, professor José Antônio G. Miranda, professor José B. Gomes and professor David Francisco M. Torres. I would like to thank my beloved wife Sabrina for always standing by my side with love, patience and care. This work couldn t have been done without you. I would also like to thank our daughter Maria Stella for being an inspiration. A very special thanks to my parents Ronaldo and Jussara for all the love, trust and support. I would like to thank my sister Clarisse for the encouragement and affection. ManythankstoLourdinha,ThomazeTomforreceivingmeintheirhome and in their family. I would also like to thank my dearests friends José Victor and Juliana for all the happy hours we spent together. Thanks to my all my friends from PUC-Rio for the companionship and support. In particular, thanks to André, Felipe, Miguel, Débora, João Paulo, Ady, Alessandro Edgar and Tiane. I would like to thank professor Ricardo Sá Earp, professor Paul Schweitzer, professor Alex Lúcio Castro, professor Carlos Tomei, professor Lorenzo Díaz and professor Jairo Bochi. I would also like to thank the staff of Secretaria de Matemática. In particular to Kátia, Creuza and Orlando. I would like to thank CNPq and FAPERJ for the financial support.

6 Abstract Pacheco, Rodrigo P.; Ruggiero, Rafael O.. Destruction of Invariant Graphs by Cˆ{1,\beta} Perturbations. Rio de Janeiro, p. Tese de Doutorado Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro. According to the theory developed by Kolmogorov, Arnold and Moser in the sixties, the majority of invariant tori persists under a C 3 perturbation of a integrable Hamiltonian. A natural question is if a perturbation in the C k topology, k < 3, still preserves such tori. Bangert showed that, in the C 1 topology, what happens is the opposite: there are metrics with no invariant torus arbitrarily close to any given Riemannian metric. Ruggiero extended these results to mechanical Lagrangians in the torus and showed that for Riemannian metrics this phenomenon is C 1 generic. In this work, we show that,givenǫ > 0, E RandareversibleTonelliHamiltonianH : TT 2 R, there exists β (0,1) and an ǫ perturbation H 0 of H in the C 1,β topology such that H 0 has no continuous invariant graphs. The result is achieved by explicitly exhibiting a Finsler metric, without any continuous field of minimizers, constructed after an analytic study of the Jacobi operator. Keywords Conformal Finsler Geometry; Jacobi equation; Lagrangian graphs; Euler- Lagrange equation; Conjugate points.

7 Resumo Pacheco, Rodrigo P.; Ruggiero, Rafael O.. Destruição de gráficos invariantes por perturbações Cˆ{1,\beta}. Rio de Janeiro, p. Tese de Doutorado Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro. Segundo a teoria desenvolvida por Kolmogorov, Arnold e Moser na década de sessenta, a grande maioria dos toros invariantes persistem após uma perturbação C 3 de um Hamiltoniano integrável. Uma pergunta natural é se perturbações em topologias C k, para k < 3, ainda preservam tais toros. Bangert mostrou que a situação é a oposta na topologia C 1 : arbitrariamente próximo de uma métrica Riemanniana plana no toro existem métricas sem nenhum toro invariante. Ruggiero estendeu esses resultados para Lagrangeanos mecânicos no toro e mostrou que, no caso de métricas Riemannianas,essefenômenoéC 1 genérico.nestetrabalho,mostramosque, dado ǫ > 0, E R e um Hamiltoniano de Tonelli reversível H : TT 2 R, existe β (0,1) e uma ǫ perturbação H 0 de H tal que H 0 não possui gráficos contínuos invariantes. Para tal, construimos explicitamente uma métrica Finsler, sem nenhum campo contínuo de minimizantes, através de um estudo analítico do operador de Jacobi. Palavras chave Geometria Finsler conforme; Equação de Jacobi; Gráficos Lagrageanos; Equação de Euler-Lagrange; Pontos conjugados.

8 Contents 1 Introduction 10 2 A geometric approach Perturbations by cones Proof of theorem Preliminaries Lagrangian viewpoint Geometrical viewpoint On Lagrangian graphs 29 4 Some ODE results Proof of proposition Conformal Jacobi equation Fermi coordinates 46 6 Construction of conformal radial perturbations Building the σ function 50 7 Proof of the main theorem Proof of theorem Appendix The Hamiltonian Jacobi equation The perturbed Jacobi equation Proof of the Hamiltonian version of the main theorem 63 Bibliography 65

9 List of Figures 2.1 In the right, the curve connecting x 1 and x 2 is a minimizing geodesic of length 2ρ that not intersect V p Ontheinterval[ r a,r a ]weusedafunctionoforderx 1 3 tosmooth f a u 1 (t) = ȳ(0) ρ (t+ρ) and u 2(t) = t+ρ x cannot intersect ŷ on (0,ρ), so there is t 0 (0,ρ) where x(t 0 ) = 0. 41