Generalizing work of Poizat in the stable case, motivated by the definability status of important spaces of definable types in algebraically closed valued fields, Cubides Kovacsics, Ye and I introduced and studied beautiful pairs for unstable theories, as a more semantic way to approach such definability questions. As one of the main applications, we could establish the strict pro-definability of all definable types of various kinds. In the talk, I will explain the main ideas behind this approach, and I will then report on some ongoing project, joint with Hrushovski, Ye and Zou, on versions of the results where a non-standard Frobenius automorphism is added to the structure. We obtain in particular surprisingly simple axioms for the resulting pairs, as these turn out to be beautiful as well. I will focus on the situation without valuation, which corresponds to the theory of proper pairs of existentially closed difference fields, where the small one is transformally algebraically closed in the large one.