Local Gibbs–Markov–Young structures for non-invertible systems

ABSTRACT

We study non-uniformly expanding systems on a compact Riemannian manifold admitting critical sets. Under some general conditions, we construct a Gibbs–Markov–Young structure on a disk whose centre's preimages are dense in the manifold. The result has the following application: in a previous study, the authors showed that the decay of correlations implies the existence of tower structure whose return time decays at the same rate. However, for technical reasons, they have to assume that the density function for the absolutely continuous measure is bounded away from 0. Now we remove this constraint and provide the arguments for the more general results.

Keywords:

• Gibbs–Markov–Young structure
• decay of correlations
• large deviation

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 37A05
• 37C40
• 37D25

Acknowledgments

The author acknowledges the referees for valuable comments and references. Also the author thanks Prof. Alves for fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Azuma-Hoeffding: Let be a sequence of martingale differences. If there is a > 0 such that ‖X_i‖_∞ < a for all 1 ≤ i ≤ n, then for all we have