Backward volume contraction for endomorphisms with eventual volume expansion

TL;DR

This paper proves that smooth maps with eventual volume expansion on compact manifolds exhibit uniform backward volume contraction on pre-orbits for almost all points, revealing a new dynamical property under mild conditions.

Contribution

It establishes a link between eventual volume expansion and backward volume contraction for almost all points in smooth dynamical systems.

Findings

Backward volume contraction holds Lebesgue almost everywhere.

Results apply to smooth maps with mild conditions of volume expansion.

Provides insights into the structure of pre-orbits in dynamical systems.

Abstract

We consider smooth maps on compact Riemannian manifolds. We prove that under some mild condition of eventual volume expansion Lebesgue almost everywhere we have uniform backward volume contraction on every pre-orbit of Lebesgue almost every point.