What I already know

Before I ask my question, I would prefer to briefly explain what I already know, so that any gap in my understanding could be rectified.

Note: I consider only bounded phase space and bounded motion.

Properties of space-filling curves

  • There can be no continuous bijection from the unit interval $[0,1]$ to the unit square $[0,1]^2$. All space-filling curves (which are continuous by definition) filling the unit square must therefore either have self-crossings or have to pass through the same point multiple times (like the Peano Curve and Hilbert Curve).

  • Any curve filling the unit square must have Hausdorff dimension 2 – as the sqare has Hausdorff dimension 2. This is indeed true for the Hilbert Curve and Peano Curve.

  • One can draw a continuous curve that has Hausdorff dimension 2 but neither self-crosses nor passes through any point multiple times, but such a curve will not fill up the unit square (or any other convex set) completely. Osgood Curves are of this nature.

  • Non-continuous curves can definitely fill the unit square without self-crossing anywhere or passing any point multiple times.

Trajectories in phase space

  • If we have an integrable system obeying Hamiltonian dynamics, then we could always, in principle, convert such a system into action–angle variables, which makes $N$ of the ‘action’ variables constants of motion, leaving behind $N$ ‘angle’ variables.

  • Now if we have bounded motion (say for example, periodic oscillations of $N$-dimensional isotropic harmonic oscillator), then the angle variables are literal angles, restricting each of them to lie in the interval $[0,2\pi]$.

  • So, having $N$ of such angle variables restricts the trajectory into an $N$-dimensional torus ${\mathbb T}^N$. Thus, if we divide the $2N$-dimensional phase space into ‘blocks’ of ${\mathbb T}^N$, the trajectory lies on one particular toroidal block.

Commensurate non-chaotic trajectories

  • Now, if we consider an $N$-dimensional anisotropic uncoupled oscillator $V(x_1,\ldots,x_n) = \frac{1}{2} m\omega_1^2 x_1^2 +\ldots + \frac{1}{2} m \omega_n^2 x_n^2\ $ such that the ratio of any two oscillation frequencies $\left(\omega_a/\omega_b\right)$ is rational, then the trajectory is simply a closed spiralling orbit restricted to move on one toroidal block.

  • Since the orbit is closed, the trajectory can automatically never fill up the entire toroidal surface. The system is therefore non-ergodic.

  • Since the system is non-ergodic (and non-mixing), the question of chaos doesn’t even arise here.

Incommensurate non-chaotic trajectories

  • If instead, the ratio of oscillation frequencies $\left(\omega_a/\omega_b\right)$ is irrational, then the trajectory does not close on itself anymore.

  • The trajectory comes arbitrarily close to itself, eventually passing through all the points of the surface but only once, and without any self-crossing anywhere, and fills up the entire toroidal surface. The system is therefore ergodic, but only on the torus.

  • However, even though the system is ergodic, it is so only on one toroidal block, while the relevant domain of phase space available for motion is the entire energy hypersurface (which is $2N-1$-dimensional). The system cannot leave one toroidal block and jump to another, making it impossible to explore the entire energy hypersurface.

  • Since ergodicity (and strong mixing) is essential for chaos (and also exponential sensitivity to initial conditions), these trajectories are non-chaotic.

  • Another fact that makes this orbits non-chaotic' is that the motion on the Torus is very regular, unlike chaotic trajectories which undergo ‘random’ motion.

  • Being non-chaotic trajectories, they have Hausdorff dimension 1 (same as that of a 1D line).

Chaotic trajectories

  • If we have a non-integrable system obeying Hamiltonian dynamics, then even in principle we cannot write down any analytic solutions to the equations of motion. This makes the system locally solvable and not globally, that is, we can only solve for nearby instants of time and not for arbitrary long times.

  • If we try to solve for arbitrary long times, we need infinite precision of the initial conditions in order for our solutions to accurately match with the results. If there is even a slight imprecision of initial conditions, the results divert wildly. Thus, there is an exponential sensitivity of initial conditions.

  • The system being non-integrable, there is a priori no such constraint as to move on the surface of one particular torus. It can explore the entire energy hypersurface (ergodic) and also strongly mix. (There could be some other constraints which will restrict the domain of phase space, but even in that case it will explore the relevant domain entirely and strongly mix). Thus the trajectories are chaotic.

  • Being chaotic trajectories, they have a Hausdorff dimension between $1$ and $2N-1$.


Question 1

Assuming the trajectories are continuous and keeping in mind that phase-space trajectories must not self-cross or pass through any point more than once, how can incommensurate non-chaotic trajectories fill up the entire torus if they have Hausdorff Dimension 1? They literally have zero area, zero volume, zero hypervolume and so on!

Question 2

How do the chaotic trajectories with a Hausdorff dimension that is not exactly $2N-1$, fill up the entire energy hypersurface? They take up some space indeed, but in order to fill up the entire energy hypersurface, mustn’t the space-filling chaotic curves have exactly $2N-1$ as their Hausdorff dimension?

Question 3

Even if somehow I am wrong that (in the absence of any other constraint) chaotic trajectories can fill up the entire energy hypersurface: The fact that statistical mechanics heavily depends on ergodicity implies that the phase-space trajectories of a 3D statistical system (such as, classical Maxwell–Boltzmann gas) in the $6N$-dimensional phase space must be space-filling, even if such a trajectory is non-chaotic.

For such a case, my aforesaid question remains valid. What then could be the reason for ergodicity?

Could it be possible that such trajectories are not continuous? Can curves that are not continuous and have Hausdorff dimension less than the dimension of energy hypersurface fill up the hypersurface?


1 Answer 1


I see a few crucial misunderstandings, some of which are related:

  1. The torus is not the unit square when it comes to considering space-filling curves. If we map the torus to the unit square, a curve can exit the square on the right and re-enter it on the left (at the same height). This curve is discontinuous on the unit square, but not on the torus.

    You already mentioned toroidal motion with incommensurate frequencies. Consider this motion with just two frequencies on a (2D) unit torus to get a straightforward example of how a simple curve can fill the unit torus.

  2. Hausdorff dimensions of regular dynamics:

    Being non-chaotic trajectories, they have Hausdorff dimension 1 (same as that of a 1D line).

    This is simply incorrect. I do not know what you have been told here, but a typical characterisation of chaos using dimensions would be:

    A trajectory is chaotic if its Hausdorff dimension is higher than its topological dimension.

    (There are possibly some pathologic counter-examples to this – because this applies to almost every definite statement about chaos. I am not following the progress on these.)

    What you apparently have not been told about are quasiperiodic dynamics. These are regular, non-periodic dynamics. Your incommensurate non-chaotic trajectories are the textbook example of this. For example, a regular dynamics with two incommensurate frequencies has Hausdorff dimension 2 and topological dimension 2, filling the torus.

  3. Chaos and ergodicity:

    How do the chaotic trajectories with a Hausdorff dimension that is not exactly $2N-1$, fill up the entire energy hypersurface?

    I am quite confident they don’t. It’s the nature of chaotic dynamics not to fill the entire phase space, otherwise it would be quasiperiodic.

    I am not sufficiently familiar with ergodic theory to tell you in detail which condition of what ergodic theorem are broken here or whether this example is breaking some ergodic hypothesis. However, I think the crucial factor here is that you have to consider observables (the function $f$ in your typical formulation of an ergodic theorem): While not every niche of phase space is visited, phase space is visited in such a manner that your observable doesn’t allow you to distinguish. The time average of $f$ over your chaotic trajectory is the same as the ensemble average of $f$ over all chaotic trajectories.


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