I think you are referring to Kolmogorov-Arnold-Moser (KAM) theorem. This theorem permits to have a clear definition of what a well-behaving Hamiltonian is and so, also how to define Hamiltonian chaos.
One can make things quite easy to explain by considering an integrable system undergoing the effect of a small perturbation. An integrable system will be described by a set of invariant tori. This means that a set of action-angle variables exists $(I_i,\theta_i)$ such that your system will be described by the Hamiltonian $H=\sum_i I_i\omega_i$ and your equations of motion are simply $\dot I_i=0,\ \dot\theta_i=\omega_i$. When you perturb this system, with a perturbation $\epsilon V({\bar I},{\bar\theta})$, with $\epsilon\rightarrow 0$, KAM theorem states that your tori do not change too much and, for almost all your initial conditions, the motion will happen on slightly deformed tori.
The point is that "almost all". Due to resonance conditions $\sum_im_i\omega_i=0$, you cannot grant everywhere that the motion is always bounded. There will be small volumes, small because the perturbation is small, for which the motion is not bounded and they will be covered uniformly satisfying the condition for ergodicity to hold, except for the value of the energy. These regions where the motion is not bounded are those in which chaotic motion happens. As already said, when $\epsilon\rightarrow 0$, they form a set of null measure. In this situation, the initial tori of the integrable system are still there.
When you start to increase $\epsilon$ you will see an interesting phenomenon: These regions of chaotic motion become more and more larger till the point to cover all the available phase space. All the initial tori are destroyed. In this situation the system becomes fully chaotic and the phase space is uniformly covered by the system, being motion completely unbounded, if we exclude the fact that the energy must have a definite value.
But we can increase $\epsilon$ to the point to cross toward the opposite limit $\epsilon\rightarrow\infty$. In this case, the limit of a very strong perturbation, a quite interesting phenomenon can happen: Tori reform! This could not be generic but, for the most known systems, the limit of a very strong perturbation describes again an integrable system. Chaotic motion disappears and tori reappears.
You can see an evidence of this here. You can change the intensity of the perturbation and the system will transit from regular to chaotic and back to regular motion. The system I consider here is that of a harmonic oscillator moving in a plane wave. This kind of system is interesting in studies of plasma physics. The reference is this, appeared in the Journal of Mathematical Physics.
The conclusion to be drawn from this is that a system can be fully ergodic just in a small window of the parameter space. If you want to understand the ideas underlying an ergodic behavior in a large system you have to turn your attention to quantum mechanics and to the Lieb and Simon theorem.
For a more historical overview, I have found this paper. The reason to cite it is to remember two pioneers in these lines of research.
I would add a condition
(4) If $f$ is measurable and $f(Sx)=f(x)$ almost everywhere, then $f$ is actually almost everywhere constant.
In the particular case mentioned in the OP, we are reduced to determine ergodicity of rotations on the torus (with angle $\frac ab$). And condition (4) seems to be the easier in this case.
Best Answer
I'm not sure it is along the lines you wished for but I'll comment anyhow:
The fact that the "entropy respect the ergodic decomposition" i.e. if $$ \mu =\int \mu_xd\nu(x)$$ implies $$ h_\mu (T)=\int h_{\mu_x} (T) d\nu(x)$$
(for a reference see
http://www.math.ethz.ch/~einsiedl/Pisa-Ein-Lin.pdf
section 3.5)
have consequences more or less in the form that you want. For example that if there is a unique measure of maximal entropy then it is ergodic. This quite trivial fact if used a lot.