# Yet another induction scheme for non-uniformly expanding transformations

## TL;DR

This paper presents a new induction scheme for non-uniformly expanding maps, establishing that a Gibbs-Markov-Young structure is essential for such maps to preserve an absolutely continuous measure with all positive Lyapunov exponents.

## Contribution

It introduces a novel induction scheme for non-uniformly expanding maps and proves the necessity of Gibbs-Markov-Young structures for measure preservation.

## Key findings

- Gibbs-Markov-Young structures are necessary for measure preservation in non-uniformly expanding maps.
- The new induction scheme advances understanding of the dynamics of non-uniformly expanding systems.
- The work links structural properties to measure-theoretic behavior in dynamical systems.

## Abstract

We introduce a new induction scheme for non-uniformly expanding maps $f$ of compact Riemannian manifolds, proving that the existence of a Gibbs-Markov-Young structure is a necessary condition for $f$ to preserve an absolutely continuous probability with all its Lyapunov exponents positive.