A motivic version of the theorem of Fontaine and Wintenberger

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field K of mixed characteristic and over the associated (tilted) perfectoid field Kb of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of K and Kb are isomorphic.