Unstable sets embedded in a chaotic attractor

[Decimal]

I am having a hard time understanding the discussion of chaotic sets on invariant manifolds as given in Chaos in Dynamical Systems by Edward Ott.

If the invariant manifold of a particular system contains a chaotic attractor $A$, then the transverse Lyapunov exponent $h$ will generally depend on the intial condition $\mathbf{x}_0$, with $\mathbf{x_0}$ an arbitrary point on the invariant manifold. They then state that one can typically identify the following two distinct exponents, $$\hat{h} = \mathrm{max}_{\mathbf{x_0}\in A}\left[h(\mathbf{x}_0 )\right],$$ $$h_* = h(\mathbf{x}_0) \quad \mathrm{for} \quad \mathbf{x}_0 \in A \ \ \mathrm{typical}.$$

In other words $h_*$ corresponds to the set of $\mathbf{x}_0$ that are typical with respect to the natural measure, and $\hat{h}$ corresponds with the set of $\mathbf{x}_0$ that maximize $h$. Now the book states the following,

"Thus, if one were to 'close one's eyes and put one's finger down randomly at a point on the invariant manifold' then the maximal transverse Lyapunov exponent generated by following the orbit from this point would be $h_*$."

To me this statement seems to suggest that all points on the manifold are indeed typical with respect to the natural measure. Later however the book states that those points on the manifold that are not typical, and thus do not produce a Lyapunov exponent $h_*$ are interesting since they correspond with unstable orbits. This is where I get confused, are these points still on the manifold even though the set containing them has natural measure zero? I feel like my confusion stems from me not fully grasping the concept of a natural measure yet, so any help there would be greatly appreciated.

 

[jvicens]

I can understand your confusion regarding the concept of natural measure and how it relates to the points on the invariant manifold. Let me try to provide some clarification on this topic.

Firstly, the natural measure is a way of quantifying the probability of a point being chosen randomly on the invariant manifold. This measure takes into account the dynamics of the system and is often used in chaotic systems to identify the regions of the manifold that are most likely to be visited by the system.

Now, to address your question, the points on the manifold that are not typical with respect to the natural measure still belong to the manifold. However, their probability of being chosen randomly is very low, as the set containing these points has a natural measure of zero. This means that these points are not likely to be visited by the system.

However, as the book mentions, these points are still interesting as they correspond to unstable orbits. This means that even though the system is not likely to visit these points, if it does, it will quickly move away from them due to the instability of the orbit. This is why these points are mentioned separately and not considered as part of the set of typical points on the manifold.

I hope this helps to clarify your confusion. If you have any further questions, please feel free to ask. it is important to have a clear understanding of the concepts we are working with. Keep exploring and learning, and I am sure you will grasp the concept of natural measure soon.