Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics
Artur O. Lopes Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1155-1174 doi: 10.3934/dcds.2011.29.1155
We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.
keywords: twisted Brownian motion. large deviation Geodesic flow Aubry-Mather measure
On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels
Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 661-679 doi: 10.3934/dcdsb.2008.10.661
We show that the set of $C^{\infty}$ mechanical Lagrangians $L(p,v)$ in $T^{2}$ without continuous invariant graphs in all supercritical energy levels is dense in the $C^{1}$ topology. A mechanical Lagrangian $L: T$$T^{2} \rightarrow \mathbb R$ is a function in the tangent space of the torus $T$$T^{2}$ given by $L(p,v)=\frac{1}{2}g(v,v)-U(p)$, where $g$ is a Riemannian metric and $U$ is a smooth potential.
keywords: Mather set. critical level mechanical Lagrangian Invariant graph
Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry
Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2006, 14(2): 365-383 doi: 10.3934/dcds.2006.14.365
We extend some previous results concerning the relationship between weak stability properties of the geodesic flow of manifolds without conjugate points and the global geometry of the manifold. We focus on the study of geodesic flows of compact manifolds without conjugate points satisfying either the shadowing property or topological stability, and we prove for three dimensional manifolds that under these assumptions the fundamental groups of certain quasi-convex manifolds have the Preissmann's property. This result generalizes a similar one obtained for manifolds with bounded asymptote.
keywords: hyperbolic spaces topological stability homoclinic orbits. Conjugate points Preissmann’s property shadowing property
On C1, β density of metrics without invariant graphs
Rodrigo P. Pacheco Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2018, 38(1): 247-261 doi: 10.3934/dcds.2018012

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
Cohomology and subcohomology problems for expansive, non Anosov geodesic flows
Artur O. Lopes Vladimir A. Rosas Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2007, 17(2): 403-422 doi: 10.3934/dcds.2007.17.403
We show that there are examples of expansive, non-Anosov geodesic flows of compact surfaces with non-positive curvature, where the Livsic Theorem holds in its classical (continuous, Hölder) version. We also show that such flows have continuous subaction functions associated to Hölder continuous observables.
keywords: action functions Livsic Theorem subaction functions Expansive geodesic flow cohomology of dynamical systems.

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