Mixing with piecewise isometries on a hemispherical shell

We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set E, and a particular subset of E, E₊, provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of E₊ to demonstrate their connection. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of E₊ defines fundamental aspects of mixing by cutting and shuffling.