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We 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 H{o}lder 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.
In the context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. The t
We consider a robust class of random non-uniformly expanding local homeomorphisms and Holder continuous potentials with small variation. For each element of this class we develop the Thermodynamical Formalism and prove the existence and uniqueness of
In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence
This work builds the connection between the regularity theory of optimal transportation map, Monge-Amp`{e}re equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. Accordin
For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables,