Odomutants and flexibility results for quantitative orbit equivalence

We introduce new systems that we call odomutants, built by distorting the orbits of an odometer. We use these transformations for flexibility results in quantitative orbit equivalence. It follows from the work of Kerr and Li that if the cocycles of an orbit equivalence are $\log$-integrable, the entropy is preserved. Although entropy is also an invariant of even Kakutani equivalence, we prove that this relation and $L^{\frac{1}{2}}$ orbit equivalence are not the same, using a non-loosely Bernoulli system of Feldman which is an odomutant. We also show that Kerr and Li's result on preservation of entropy is optimal, namely we find odomutants of all positive entropies orbit equivalent to an odometer, with almost $\log$-integrable cocycles. We actually build a strong orbit equivalence between uniquely ergodic Cantor minimal homeomorphisms, so our result is a refinement of a famous theorem of Boyle and Handelman. We finally prove that Belinskaya's theorem is optimal for all the odometers, namely for every odometer, we find a odomutant which is almost-integrably orbit equivalent to it but not flip-conjugate. This yields an extension of a theorem by Carderi, Joseph, Le Ma\^itre and Tessera.