# American Institute of Mathematical Sciences

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

* 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.

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
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##### References:
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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