Example for exponential growth of complexity in a finite horizon multi-dimensional dispersing billiard

We present an example for a periodic point in a three-dimensional dispersing billiard configuration around which the complexity of singularities grows exponentially with the iteration of the map. This implies the existence of multi-dimensional dispersing billiard configurations with finite horizon where the complexity grows exponentially. We also show that complexity growth can be faster than the minimum expansion of unstable vectors—a phenomenon with far-reaching consequences in the study of mixing properties for these systems. The observed behaviour is in strong contrast with the behaviour of two-dimensional systems