Dynamical systems defined by actions of Rk

• Grant number: 09/17493-4
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2010
• End date: March 31, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Carlos Alberto Maquera Apaza
• Grantee: Carlos Alberto Maquera Apaza
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The aim of this project of research is to study systematicly some aspects of dynamical systems defined by actions of R^k, k>1, more precisely:A.- we will give continuity to the study of codimension one Anosov actions of R^k that we initiate with T. Barbot in 2008. More precisely, we try to generalize results known on the classification of codimension one Anosov flows (actions of R) for actions of R^k. These results will be fundamental steps for to show, in the long term, the equivalent for actions of R^k of the well know Verjovsky's Conjecture:"codimension one Anosov flows on closed manifolds of dimension greater than 3 are topologically equivalent to a suspension of a hyperbolic toral automorphism".B.- we try to extend and to study some aspects of the ergodic theory for Anosov actions of R^k and, later for more general groups. Our approach will be the "thermodynamic formalism" for these actions. In fact, this theory is well developed for ctions of discrete groups, but to go on to the continuous case (even more, for groups with dimension > 1) has some subtleties. However, we try to add topological and geometrical methods in the study of the thermodynamic formalism, as was made in the previous works of the participants of this project. C.- to find conditions for guarantee the existence of orbits diffeomorphic to S^1 for an action of R^2 on a closed 3-manifold, that is, the existence of periodic orbits that are common for two commutative flows which generate the action. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BIASI, CARLOS; MAQUERA, CARLOS. A NOTE ON OPEN 3-MANIFOLDS SUPPORTING FOLIATIONS BY PLANES. Proceedings of the American Mathematical Society, v. 140, n. 3, p. 961-969, MAR 2012. (09/17493-4)

MAQUERA, CARLOS; VENATO-SANTOS, JEAN. Foliations and global injectivity in R-n. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 44, n. 2, p. 273-284, JUN 2013. (09/17493-4)