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January  2018, 38(1): 247-261. doi: 10.3934/dcds.2018012

On C1, β density of metrics without invariant graphs

Rodrigo P. Pacheco 1, and  Rafael O. Ruggiero 2,,

1. 

Departamento de Geometria e Representação Gráfica, IME-UERJ, R. São Francisco Xavier, 524, Rio de Janeiro, 20550-900, Brazil
2. 

Departamento de Matemática PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro 22543-900, Brazil, and Université d'Aix Marseille, France

* Corresponding author: Rafael O. Ruggiero

Received  July 2016 Revised  July 2017 Published  September 2017

Fund Project: Partially supported by FAPERJ, CNPq, CAPES and CNRS, unit´e FR2291 FRUMAM.

We show that given any $C^{\infty}$ Riemannian structure $(T^{2},g)$ in the two torus, $\epsilon >0$ and $\beta \in (0,\frac{1}{3})$, there exists a $C^{\infty}$ Riemannian metric $\bar{g}$ with no continuous Lagrangian invariant graphs that is $\epsilon$-$C^{1,\beta}$ close to $g$. The main idea of the proof is inspired in the work of V. Bangert who introduced caps from smoothed cone type $C^{1}$ small perturbations of metrics with non-positive curvature to get conjugate points. Our new contribution to the subject is to show that positive curvature cone type small perturbations are ``less singular" than non-positive curvature cone type perturbations. Positive curvature geometry allows us to get better estimates for the variation of the $C^{1}$ norm of the singular cone in a neighborhood of its vertex.

Keywords: Lagrangian graphs, conjugate points, variational calculus, geodesic flows, local perturbations.
Mathematics Subject Classification: Primary:37J30, 53B99;Secondary:37J40.
Citation: Rodrigo P. Pacheco, Rafael O. Ruggiero. On C1, β density of metrics without invariant graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 247-261. doi: 10.3934/dcds.2018012
References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-56.   Google Scholar

[2]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de lánneau Vol. 1, Astérisque, (1983), 103-104.   Google Scholar

[3]

M. R. Herman, Non existence of Lagrangian graphs, available online in Archive Michel Herman: http://www.college-de-france.fr, (1990), 1–5. Google Scholar

[4]

R. S. Mackay, A criterion for non-existence of invariant tori for Hamiltonian systems, Physica D: Nonlinear Phenomena, 36 (1989), 64-82.  doi: 10.1016/0167-2789(89)90248-0.  Google Scholar

[5]

J. N. Mather, Destruction of invariant circles, Ergodic Theory Dynam. Systems, 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.  Google Scholar

[6]

M. Morse, The Calculus of Variations in the Large American Mathematical Society, Providence, RI, 1996. doi: 10.1090/coll/018.  Google Scholar

[7]

G. Paternain, Geodesic Flows Progress in Mathematics, 180, Birkhäuser Boston, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[8]

R. O. Ruggiero, On the creation of conjugate points, Mathematische Zeitschrift, 208 (1991), 41-55.  doi: 10.1007/BF02571508.  Google Scholar

[9]

R. O. Ruggiero, The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the C1 topology, Bulletin of the Brazilian Mathematical Society, 35 (2004), 377-385.  doi: 10.1007/s00574-004-0020-0.  Google Scholar

[10]

R. O. Ruggiero, On the density of mechanical Lagrangians in T2 without continuous invariant graphs in all supercritical energy levels, Discrete and Continuous Dynamical Systems. Series B, 10 (2008), 661-679.  doi: 10.3934/dcdsb.2008.10.661.  Google Scholar

[11]

F. Takens, A C1 counterexample to Moser's twist theorem, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., 33 (1971), 379-386.   Google Scholar

show all references

References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-56.   Google Scholar

[2]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de lánneau Vol. 1, Astérisque, (1983), 103-104.   Google Scholar

[3]

M. R. Herman, Non existence of Lagrangian graphs, available online in Archive Michel Herman: http://www.college-de-france.fr, (1990), 1–5. Google Scholar

[4]

R. S. Mackay, A criterion for non-existence of invariant tori for Hamiltonian systems, Physica D: Nonlinear Phenomena, 36 (1989), 64-82.  doi: 10.1016/0167-2789(89)90248-0.  Google Scholar

[5]

J. N. Mather, Destruction of invariant circles, Ergodic Theory Dynam. Systems, 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.  Google Scholar

[6]

M. Morse, The Calculus of Variations in the Large American Mathematical Society, Providence, RI, 1996. doi: 10.1090/coll/018.  Google Scholar

[7]

G. Paternain, Geodesic Flows Progress in Mathematics, 180, Birkhäuser Boston, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[8]

R. O. Ruggiero, On the creation of conjugate points, Mathematische Zeitschrift, 208 (1991), 41-55.  doi: 10.1007/BF02571508.  Google Scholar

[9]

R. O. Ruggiero, The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the C1 topology, Bulletin of the Brazilian Mathematical Society, 35 (2004), 377-385.  doi: 10.1007/s00574-004-0020-0.  Google Scholar

[10]

R. O. Ruggiero, On the density of mechanical Lagrangians in T2 without continuous invariant graphs in all supercritical energy levels, Discrete and Continuous Dynamical Systems. Series B, 10 (2008), 661-679.  doi: 10.3934/dcdsb.2008.10.661.  Google Scholar

[11]

F. Takens, A C1 counterexample to Moser's twist theorem, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., 33 (1971), 379-386.   Google Scholar

Figure 1.  In the right, the curve connecting $x_1$ and $x_2$ is a minimizing geodesic of length $2\rho$ that not intersect a neighborhood of the singularity
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