1 - Lyapunov functions, strongly homogeneous sets and star flows (preprint 2016)
LUCIANA SALGADO
ABSTRACT: We say that a flow or vector field X C^1 is star on a compact invariant set Λ if
there exist neighborhoods V of X and U of Λ in M for which every closed orbit in U of
every vector field Y in V is hyperbolic. In this work, we present a characterization of a flow to be
star. It is obtained conditions to strong homogeneity for singular sets for a C^1 flow by using the
notion of infinitesimal Lyapunov functions. As an application we obtain some results related to
singular hyperbolic sets for flows.
2 - SINGULAR HYPERBOLICITY AND SECTIONAL LYAPUNOV EXPONENTS OF VARIOUS ORDERS (preprint https://arxiv.org/pdf/1611.04072v2.pdf)
LUCIANA SALGADO
ABSTRACT. It is given notions of singular hyperbolicity and of sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic set. It is obtained a characterization of singular hyperbolcity in this broad sense, by using the notion of infinitesimal Lyapunov functions. Furthermore, it is reduced the requirements to obtain singular hyperbolicity. As an application we obtain some results related to singular hyperbolic sets for flows.