Good inducing schemes for uniformly hyperbolic flows,
and applications to exponential decay of correlations
Ann. Henri Poincaré 26 (2025) 921-945.
Ian Melbourne and Paulo Varandas
Abstract
Given an Axiom A attractor for a C1+α flow (α>0), we construct a countable Markov extension with exponential return times in such a way that the inducing set is a smoothly embedded unstable disk. This avoids technical issues concerning irregularity of boundaries of Markov partition elements and enables an elementary approach to certain questions involving exponential decay of correlations for SRB measures.
pdf file,
link to journal version
Typos:
There are some minor inaccuracies in the proof of Lemma 2.9:
To ensure disjointness, δ0 in claim (**) should be
δ0/2. So in the statement of the lemma, need to choose
ε<δ0/4. Should also shrink δ (second line of proof) so that
C3(3δ)α<δ0/4.
When applying (2.2), the constant C3 should be C2.
Here is a file showing the changes.
In Proposition 2.4, we tried to clarify an argument that seemed to be missing in the literature. But the proof of Proposition 2.4 is also unclear.
It turns out that there is a simple solution: make the property of ULnj claimed in Proposition 2.4 part of the definition of ULnj in the first place.
It is easily checked that this raises no issues for the remainder of the proof of Theorem 2.1. (It may be that this is what was intended in the literature anyway.)
A clean proof (with the simplified definition) of Theorem 2.1 in the simplest setting of uniformly expanding maps is written out in our unpublished notes.