Robust Wada Boundaries and Entropy Scaling in pp-Wave Spacetimes

We study the dynamics of the geodesics of pp-wave spacetimes with polynomial profiles, which are dynamically equivalent to the motion of a classical particle in a two-dimensional harmonic polynomial potential. We demonstrate that the Wada property of the escape basins is robust under variation of the polynomial degree, i.e., the basin boundaries remain maximally intermingled as the number of escape channels increases. We further provide a quantitative characterization of the degree of dynamical uncertainty by computing the basin entropy $S_{b}$ and the boundary basin entropy $S_{bb}$. We find that these measures increase monotonically with the polynomial degree, indicating enhanced unpredictability of the final state of the system. We also show that $S_{bb}$ is greater than $\ln(2)$ for $n>3$, and this confirms that the basin boundaries are fractal.