{'fraction_non_alphanumeric': 0.018819503849443968, 'fraction_numerical': 0.0, 'mean_word_length': 5.0, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards

When considering hyperbolicity in multi-dimensional Hamiltonian sytems, 
especially in higher dimensional billiards, the literature usually 
distinguishes between dispersing and defocusing mechanisms. In this paper 
we give a unified treatment of these two phenomena, which also covers the 
important case when the two mechanisms mix. Two theorems on the hyperbolicity 
(i.e. non-vanishing of the Lyapunov exponents) are proven that are hoped to 
be applicable to a variety of situations. 
 
   As an application we investigate soft billiards, that is, replace the hard 
core collision in dispersing billiards with disjoint spherical scatterers 
by motion in some spherically symmetric potential. Analogous systems in two 
dimensions have been widely investigated in the literature, however, we are 
not aware of any mathematical result in this multi-dimensional case. 
Hyperbolicity is proven under suitable conditions on the potential. This 
way we give a natural generalization of the hyperbolicity results 
obtained before in two dimensions for a large class of potentials.