We study a stroboscopic quantum Ising model with Fibonacci dynamics. We use its boundary spin correlation functions in long but finite chains to identify regions in the phase diagram which exhibit Majorana zero modes (MZM) as well as Majorana golden-ratio modes (MGM). We find that these regions evolve in a self-similar manner with increasing simulation time and identify the self-similarity transform which governs this evolution of the phase diagram. Integrability-breaking perturbations lead to a temporal decay of the boundary spin correlations, ultimately limiting the self-similarity of the phase diagram. Our predictions are testable with current quantum information processors.
