On the volume of sectional-hyperbolic sets

TL;DR

This work addresses the volume properties of sectional-hyperbolic sets for flows on a $d$-dimensional manifold ($d\ge 3$). By combining partial hyperbolicity with a graph-transform method, it shows that a transitive sectional-hyperbolic set of positive volume must coincide with the entire manifold, and in fact be uniformly hyperbolic without singularities. The argument uses alpha-limit techniques and invariant-manifold structure to upgrade sectional expansion to full hyperbolicity and to deduce that the set is both stable and unstable saturated, hence equal to $M$. The results extend known zero-volume findings in three dimensions to higher dimensions, linking volume conditions to global hyperbolicity and yielding strong dynamical consequences for sectional-hyperbolic systems.

Abstract

For a transitive sectional-hypebolic set $Λ$ with positive volume on a $d$-dimensional manifold $M$($d\ge3$), we show that $Λ=M$ and $Λ$ is a uniformly hyperbolic set without singularities

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark
• Lemma 3.1
• Lemma 3.2
• Remark
• Lemma 3.3