# On the volume of sectional-hyperbolic sets

## TL;DR

This paper proves that any transitive sectional-hyperbolic set with positive volume on a manifold must be the entire manifold and is uniformly hyperbolic without singularities.

## Contribution

It establishes a rigidity result linking positive volume, transitivity, and uniform hyperbolicity for sectional-hyperbolic sets.

## Key findings

- Any such set with positive volume equals the whole manifold.
- The set is uniformly hyperbolic and contains no singularities.
- This characterizes the structure of sectional-hyperbolic sets with positive volume.

## Abstract

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