An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

v3: The published article (Ergodic Theory Dynam. Systems 40, 2020) contained a significant error in Lemma 2.14, used in the core Theorem 4.1. This is a consolidated version of the article, with the error corrected (and a few other minor points improved along the way). In order to fix the error, the assumption on coupling in 4.1 needs to be slightly modified (see also Definition 2.12) and Lemma 2.14 (now numbered 2.15) has been adjusted. Section 5.1 provides a criterion to ensure this new hypothesis in our cases of interest, so that all other results are unaffected. I apologize to readers of the previous version for this embarrassing mistake, and warmly thank Manuel Stadlbauer for pointing out this error to me.International audienceWe consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials (" flatness ") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer operator of the same speed than entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of Hölder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situation, including Pomeau-Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results