Topological dynamics and NIP fields

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field $K$. We show that the Ellis group of the universal definable flow of $\mathrm{SL}_2(K)$ is non-trivial if the multiplicative group of $K$ is not type-definably connected, providing a way to find multiple counterexamples to the Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over $K$ with definable f-generics.