Por Luciana Silva Salgado (Universidade Federal do Rio de Janeiro).

The study of hyperbolic structures (uniform and nonuniform ones) is a central subject in Dynamical Systems.

Nowadays, there are many notions of weak hyperbolicity, and here I am interested in the setting of flows with singularities (e.g., Lorenz systems). In this talk I am going to talk about some notions of (nonuniform) sectional hyperbolicity (in the sense of p-planes expansion) for C1  flows.

And, how to use of the powerful tool of quadratic forms (Lyapunov Functions) to characterize dynamical properties and to obtain ergodic features for those kind of systems.
Finally, if time permits, I will state some new result involving SRB measures for it, in a jointly work with V. Araujo (UFBA, Brazil) and Sergio Sousa (UFRJ, Brazil).

References:

• [1] V. Araujo, L. Salgado. Infinitesimal Lyapunov functions for singular flows. Math. Z., 275, 3-4, 863–897. 2013.
• [2] A. Arbieto, L. Salgado. On critical orbits and sectional hyperbolicity of the nonwandering set for flows. J. of diff. equations 250 (2011) 2927–2939.
• [3] V. Araujo, L. Salgado, S. Sousa. Physical measures for mostly sectional expanding flows. ArXiv:2205.04207, 2023.
• [4] L. Salgado. Singular hyperbolicity and sectional Lyapunov exponents of various orders. Proc. of the Amer. Math. Soc., Vol. 147, doi.org/10.1090/proc/14254. 2019.