Chaos and ergodicity in hamiltonian systems EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be globally ergodic.
So the new formulation of my question is now this : 
Do we call chaotic any hamiltonian system that exhibits the usual chaotic properties on each hypersurface of constant energy (e.g. ergodicity, mixing, positive entropy, positive Lyapunov exponent, etc.) or do we require a "complicated" geometry of those hypersurfaces ?
For example, imagine a system whose hypersurfaces of constant energy are very simple, like planes $z$=constant, but with complicated, chaotic behaviour on each of those planes. Would you call that system chaotic ? Thank you for your thoughts !

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*Yep: Alfredo M. Ozorio de Almeida wrote about this: 
http://books.google.co.uk/books?id=nNeNSEJUEHUC&pg=PA60&lpg=PA60&dq=hamiltonian+chaos+liouville+measure&source=bl&ots=63Wnmn-xvT&sig=Z0eRtIQxmdQvgWUcLBab7ZJ9y-U&hl=en&ei=0EXfTuvzJcOG8gP5mZjaBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAw#v=onepage&q=hamiltonian%20chaos%20liouville%20measure&f=false

*What is meant by Chaos:
Laplace said, standing on Newton's shoulders, "Tell me the force and where we are, and I will predict the future!" An elusive claim, which assumes the absence of deterministic chaos:
Deterministic time evolution does not guarantee predictability, which is particularly relevant for mechanical systems whose equations are non-integrable - common in systems which have nonlinear differential equations with three of more variables.
Knowing this, we enter the realm of physics:
In the hamiltonian formulation of classical dynamics, a system is described by a pair of first-order ordinary equations for each degree of freedom, so in addition we re-impose the conditions from deterministic time evolution (the nonlinear differential equations) and given the space is constrained: Ergodicity!
Hamiltonian Chaos: http://www.phys.uri.edu/~gerhard/hamchaos.html
