Impact of chaos on a system's entropy

[L_landau]

From a heuristic standpoint it makes sense that when a system goes from being periodic to chaotic, the occupied volume of the phase space increases (while not violating liouville theorem). Since the volume of phase space is proportional if not equal to the entropy, shouldn’t entropy always increase when a system becomes chaotic? Are there counter examples to this?

 

[anorlunda]

Are you asking about fine-grained entropy or coarse-grained entropy?