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.