Singular Hyperbolicity and sectional Lyapunov exponents of various orders

TL;DR

This paper extends the concepts of singular hyperbolicity and sectional Lyapunov exponents to higher orders, providing new characterizations and criteria for dominated splittings and hyperbolic structures in dynamical systems.

Contribution

It introduces generalized notions of singular hyperbolicity and sectional Lyapunov exponents beyond classical dimensions, with new characterizations and alternative conditions for hyperbolic behavior.

Findings

Characterization of dominated splittings using Lyapunov exponents

New criteria for singular hyperbolicity

Results on singular hyperbolic sets for flows

Abstract

It is given notions of singular hyperbolicity and 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 dominated splittings, partial and singular hyperbolicity in this broad sense, by using Lyapunov exponents and the notion of infinitesimal Lyapunov functions. Furthermore, it is given alternative requirements to obtain singular hyperbolicity. As an application we obtain some results related to singular hyperbolic sets for flows.