What is the meaning of volume of dynamical system

Question

https://en.wikipedia.org/wiki/Dynamical_system explains that the volume of state space or phase space is invariant. The lecture note titled "11 Strange attractors and Lyapunov dim." taken from the book of Strogatz shows in eq(2) a coordinate transformation of the volume. I want to understand if the proof shown in the note means that the volume of systems with strange attractor is invariant under some kind of transformation such as the coordinate transformation. By coordinate transformation, we can generate the phase space reconstruction and using that we can get a strange attractor. Upon proper choice of parameter setting for the chaotic dynamical system we can see the strange. But I am unable to understand the proof.

Question: Can somebody please show how to prove that the volume for systems having strange attractors is invariant to transformation and what this means.

Does the volume of strange attractors shrink or expand?

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UPDATE: 18 Aug

Based on the discussion under the comments, this is what I could write from whatever I could understand. Shall appreciate help in finishing the proof writing in an elegant way.

Proof: volume of strange attractor shown by systems in chaotic dynamics is invariant under some transformation and is a measure or a metric.

My idea is that, let $n_a$ be the attractor dimension and $d$ be the embedding dimension and the attractor has a volume $v$ with an attractor dimension $n_a$. If scalar valued time series are available, then we can reconstruct the attractor in $d$ dimensional phase space by Takens' delay embedding method, $d \ge 2n+1$ where $n$ is the dimension of the observed system. We don't hae knowledge of the actual value of $n_a$. Since, for dissipative systems volume $v \le 0$, if and only if $n \le n_a$, and is equal to zero since its dimension is less than $n_a$. Therefore any dissipative system preserves the volume of the attractor, which is zero. As for the change of coordinates, since the attractor is a measure zero set, the image of the attractor under any smooth map will also be measure zero.

Now how do I prove that the attractor is a measure set zero and is a metric like the Lebesgue measure? Can somebody please help in formally writing this proof? Thank you.

Answer 1

When they say volume, they really mean `measure.' A measure on a space $X$ is a function $\mu$ that assigns lengths (or areas, or volumes, or probabilities--the specific space $X$ or the context usually dictates how you think of what the measure is, well, measuring) to "nice" subjects of $X,$ where "nice" means beforehand someone selected some subsets of $X$ that we can measure. These are called the measureable sets.

A map $T : X\rightarrow X$ is said to be $\mu$-invariant if (a) whenever $S$ is measureable, so is $T^{-1}(S)$, and (b) $\mu(T^{-1}S) = \mu(S)$ whenever $S$ is measureable.

As to how to check it, this depends a lot on the particulars. One incredibly common, helpful trick is that you don't need to check that conditions (a) or (b) holds for every measureable subset--if you check (a) and (b) on a family of sets that `generates' the collection of measureable sets, then you can conclude it holds everywhere. For instance, if your space was $X = [0, 1]$ with the usual "Lebesgue measure" assigning a subset of $X$ it's length, it'd suffice to check that $T$ preserves measures of intervals.

Answer 2

A couple things:

1. Notice the note under equation 2:

Dissipative systems have attractors, while volume conserving systems cannot have attractors nor repellers.

This is true in the sense where "volume" means the Lebesgue measure, i.e., the normal definition of volume on $\mathbb{R}^n$. Attractors are necessarily of a lower dimension than the phase space itself, so its volume (in the Lebesgue sense) must be 0; e.g., the volume of a surface in $\mathbb{R}^3$ is 0 since the surface is 2-dimensional. Maybe this preservation of volume is trivial because the attractor necessarily has Lebesgue volume zero.

So this seems to answer your question on the face of it. However, dynamics on strange attractors are typically ergodic, which is the section you are reading in the first Wikipedia article. Ergodic dynamics typically have something called an invariant measure, which means that there is some notion of volume (the measure) which is preserved by the dynamics (invariant). Therefore, if one can parameterize the attractor, i.e, find a change of coordinates from $\mathbb{R}^n$ to the attractor, then "volume" in the sense of the invariant measure of the attractor and dynamics will indeed be preserved.