Deterministic many-body dynamics with multifractal response

Dynamical systems can display a plethora of ergodic and ergodicity breaking behaviors, ranging from simple periodicity to ergodicity and chaos. Here we report an unusual type of non-ergodic behavior in a many-body discrete-time dynamical system, specifically a multi-periodic response with multi-fractal distribution of equilibrium spectral weights at all rational frequencies. This phenomenon is observed in the momentum-conserving variant of the newly introduced class of the so-called parity check reversible cellular automata, which we define with respect to an arbitrary bi-partite lattice. Although the models display strong fragmentation of phase space of configurations, we demonstrate that the effect qualitatively persists within individual fragmented sectors, and even individual typical many-body trajectories. We provide detailed numerical analysis of examples on 2D (honeycomb, square) and 3D (cubic) lattices.