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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January
2018, 38(1): 247-261.
doi: 10.3934/dcds.2018012
On C1, β density of metrics without invariant graphs
| 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 |
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 and 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.
|
| [2] |
M. R. Herman,
Sur les courbes invariantes par les difféomorphismes de lánneau Vol. 1, Astérisque, (1983), 103-104.
|
| [3] |
M. R. Herman, Non existence of Lagrangian graphs, available online in Archive Michel Herman: http://www.college-de-france.fr, (1990), 1–5. |
| [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. |
| [5] |
J. N. Mather,
Destruction of invariant circles, Ergodic Theory Dynam. Systems, 8 (1988), 199-214.
doi: 10.1017/S0143385700009421. |
| [6] |
M. Morse,
The Calculus of Variations in the Large American Mathematical Society, Providence, RI, 1996.
doi: 10.1090/coll/018. |
| [7] |
G. Paternain,
Geodesic Flows Progress in Mathematics, 180, Birkhäuser Boston, 1999.
doi: 10.1007/978-1-4612-1600-1. |
| [8] |
R. O. Ruggiero,
On the creation of conjugate points, Mathematische Zeitschrift, 208 (1991), 41-55.
doi: 10.1007/BF02571508. |
| [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. |
| [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. |
| [11] |
F. Takens,
A C1 counterexample to Moser's twist theorem, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., 33 (1971), 379-386.
|
show all references
References:
| [1] |
V. Bangert,
Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-56.
|
| [2] |
M. R. Herman,
Sur les courbes invariantes par les difféomorphismes de lánneau Vol. 1, Astérisque, (1983), 103-104.
|
| [3] |
M. R. Herman, Non existence of Lagrangian graphs, available online in Archive Michel Herman: http://www.college-de-france.fr, (1990), 1–5. |
| [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. |
| [5] |
J. N. Mather,
Destruction of invariant circles, Ergodic Theory Dynam. Systems, 8 (1988), 199-214.
doi: 10.1017/S0143385700009421. |
| [6] |
M. Morse,
The Calculus of Variations in the Large American Mathematical Society, Providence, RI, 1996.
doi: 10.1090/coll/018. |
| [7] |
G. Paternain,
Geodesic Flows Progress in Mathematics, 180, Birkhäuser Boston, 1999.
doi: 10.1007/978-1-4612-1600-1. |
| [8] |
R. O. Ruggiero,
On the creation of conjugate points, Mathematische Zeitschrift, 208 (1991), 41-55.
doi: 10.1007/BF02571508. |
| [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. |
| [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. |
| [11] |
F. Takens,
A C1 counterexample to Moser's twist theorem, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., 33 (1971), 379-386.
|
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