We prove the 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 $K^{ \flat}$ 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 $K^\flat$ are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces.
Categories: math.AG math.NT