• Trevor L. Irwin and Salawomir Solecki, Projective Fraïssé limits and the pseudo-arc.

]]>A fascinating distinction for fractal-like objects! Tell me, are dragon curves more like the period-doubling examples, or more like the everywhere-difficult examples?

]]>It looks like the two time series match fairly well and so perhaps there will be no El Nino until 2019.

]]>• Erwan Lanneau, Tell me a pseudo-Anosov, *Newsletter of the European Mathematical Society* **106** (December 2017), 11–16.

Thanks for the connection of Sharkovskii order to period doubling.

Here is some applied math/physics–

From working on the ENSO model, it’s clear that a period doubling exists in the ENSO time series. Many climate researchers believe that this time-series is chaotic, almost to the point of being snake-like. Yet, put together a period-doubling (i.e. annual to biennial) amplification on top of the highly precise cyclic lunar forcing and this may deconvolute the mess. I will present these results next week at the AGU meeting. This is a fit one can achieve without too much trouble:

I dabbled a bit in the intersection of number theory

and fractals. Something i saw was a connection between

the Calkin-Wilf sequence

https://en.wikipedia.org/wiki/Calkin-Wilf_tree

and the Sharkovskii order

https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem

on the positive integers. The Calkin-Wilf sequence is

a bijection from the positive integers to the positive

rationals. The recurrence relation,

is a concise expression for generating the sequence

(). The Sharkovskii order is based on a

collection of sequences of positive integers which help

account for the structure of the standard period

doubling diagram.

For positive integers j,k the partially open rectangles

help describe symmetries in the graph of

(for k=1 set ).

Let be the intersection of the graph of

with . is

empty if j<k. The function

is a bijection from to

. Each is

invariant under the “reflections”

but an with more than 1 point is only

invariant under the “rotation”

The center of symmetry of is

. This point is

in if and only if j=k. It is the only

point in . For the

number of points in doubles as we

increment j while keeping k fixed.

This is not very surprising since the intervals

double in length as we

increment j. However the top row of rectangles,

, contains the image of the sequence

3, 5, 7, 9, 11, …

The row of rectangles below it contains the image of

the sequence

In general, for a fixed k the rectangles,

$\latex R_{j,k}$, contain the image of the sequence

This sequence of sequences defines the Sharkovskii

order on the positive integers aside from the powers of

2. The powers of 2 are mapped by q to the bottom left

vertex on the boundary of the rectangles

. These point sort of form the

bottom of the graph of q(N). And the powers of 2 form

the final sequence in the Sharkovskii order.

The Koopman operator is the one-sided inverse to the transfer operator so that but . The simplest time-irreversability discussion I know of is Fully Chaotic Maps and Broken Time Symmetry | Dean Driebe | Springer There’s actually a much simpler way of presenting his material, but, whatever. The transfer/koopman operators show up in special functions, where they relate multiplicative functions from number theory to the multiplication theorem for the gamma function, polylogarithm, Bernoulli polynomials etc. The special functions are essentially the eigenfunctions of the koopman/transfer operators that correspond to the multiplicative functions.

I once saw a “koopmanistic” lecture of the pre-quantization (expansion to first order in ) of billiards-in-a-box (ideal gas) and it was excellent and fascinating and I’ve been killing myself to find it again. The wave functions were fractals! They destructively interfere when all the billiards are on one side of the box! For things like the transfer operator of the Bernoulli map, one can “trivially” exhibit fractal eigenfunctions, so not a total surprise, but an ideal gas is a step up in sophistication.

The Moyal product of pre-quantization is a special case of the star product on universal enveloping algebras. This suggests that universal enveloping algebras have, um, fractal eigenfunctions in them, in general. You will just hate me for saying this, but, it seems that possibly, in general, operators with discrete spectra that have smooth eigenfunctions for those spectral points, also have fractal eigenfunctions (viz nowhere differentiable) with a continuous spectrum. I can show you special cases, but they’re hard to construct and hard to characterize and generalize. Connecting those dots would be interesting, but again, life is short.

]]>Disclaimer: I’ve never done astrodynamics! The simplest model I know of for phase-locking is the kicked rotor and the circle map. The laboratory-bench device consists of two disks; one is free to spin, the other is attached to a motor. A weak, stretchy spring connects to a point on each rim. You turn on the motor, and the freely-spinning disk will turn also, pulled by the spring. The speed of the freely-spinning disk will quickly lock into some integer ratio of the motor; the dominant ones being 1:1 but also 2:1, 1:4 will be strong phase-locking regions. In principle, it can phase lock to *any* integer ratio p:q, although most of these are very weak and hard to lock onto. In electronics, phase-locked loops (PLL’s) use this principle to lock onto a radio signal; they’re very widespread.

The mathematical model for this is the “circle map”, the iterated equation The value of can be thought of as the angle of the spinning disk. The constant value of can be thought of as the frequency of the driving motor. The small constant is supposed to be the weak coupling of the spring. Basically, if the freely-spinning disk races ahead, the K pulls it back. If it lags, it gets pushed forward.

There are two free parameters: the speed of the motor, the strength of the spring. What happens? Either it phase locks promptly, or its its in a region where it has trouble locking on, and bounces chaotically. The phase-locked regions are called Arnold tongues, the wikipedia article has explanations and pretty pictures (made by me). There are more here – a tall one, a close-up and a different closeup. They’re pretty.

The orbital dynamics problem would be solved by picking a set of orbital coordinates for the moons, discover that the forces between them resemble the circle map (replacing the sine by some other periodic function, as needed), and then concluding “ah hah!”. I assume this just all works out with maybe some interesting hiccups along the way.

The clearing of the dust lanes starts the same way, except that now one realizes one is also dealing with an Anosov flow. If you placed two pieces of dust near each other in a dust-free lane, two things happen. The phase-locking forces would probably push them closer to one-another in the angular direction, but pull them apart in the radial direction. This is characteristic of Anasov flows: in some directions, trajectories (geodesics) converge; in other (orthogonal) directions, they diverge. The expanding, diverging geodesic flows clear the dust-free lanes.

Traditionally, these would be called “tidal forces”, appearing in the stress tensor, but I find that name unintuitive for this particular setting: Its really the geodesic flow, the integral curves, the visual shape of tangent manifolds that is visualizable, that can be captured in a motion-picture. Telling someone “oh its just tides” is utterly unintuitive.

BTW, such flows also have some “neutral” directions, orthogonal to the stretch and compress directions. Here, this would be the direction orthogonal to the orbital plane — there are no forces in that direction.

Motion in the dusty lanes is chaotic and mixing (in the formal sense of mixing), so if you could drop a bag of brightly colored glitter into one of the dust lanes, you’d find that it spread all over the place in short order.

So the above is the “physical”, “intuitive” explanation, and again, its pure speculation on my part, but I believe its correct. Some remarks: first about topology. The Arnold tongues map a set of measure zero (the rationals) into a set of finite measure (the tongues, one distinct, unique tongue per rational). How does that work? Seems magical to me. I really don’t get it. Topology is weird.

The appearance of lanes suggests that there is some operator having some spectrum. What’s that operator? How is it related to the transfer operator (and what is the transfer operator for these problems, anyway?). (The transfer operator is what tells you about the bulk motion of dust). I’ve never, ever seen even a hint of a discussion of this, anywhere. For me, personally, it would be a good excuse to study index theorems, although they might have little relation to the problem. Developing an operator-theoretic understanding of this phenomena is what would interest me.

]]>Linas wrote:

Dynamical systems in a narrow sense happen on symplectic manifolds, which is what Abraham & Marsden explain. One can thence journey to affine geometry on fibre bundles. Or to Riemannian geometry which seems prettiest when there’s a spin structure. Which leads to Clifford algebras and tensor algebras and Hopf algebras and those pesky affine Lie algebras.

All math is connected… but I probably know about ten times more about any one of these other topics than I do about dynamical systems, so invoking them mainly reminds me that I’m more interested in all these other subjects than dynamical systems!

I could get interested in dynamical systems… but only if I had some project or hobby that would pull me in.

For example, one thing I’d like to understand is why in celestial resonance sometimes seems to *stabilize* orbits and sometimes seems to *destabilize* them. For example, the moons of Jupiter seem to ‘enjoy’ being in resonance:

but the rings of Saturn have *gaps* at orbits that are in resonance with the moons:

The most intriguing part of dynamical systems, to me, is the transfer operator. This operator replaces point dynamics (where did this trajectory/geodesic go?) by the dynamics of smooth distributions (how does the ensemble/bulk evolve?). The most shocking thing that it does is to take systems that seem time-reversible (when viewed as individual trajectories) and make them time-irreversible (when viewed as a bulk).

As you probably know, this subject is sometimes called Koopmanism. Prigogine has talked a lot about the ’emergent irreversibility’ that arises from this viewpoint. I guess a good intro is here:

• B. Misra, I. Prigogine and M. Courbage, From deterministic dynamics to probabilistic descriptions, *Physica A: Statistical Mechanics and its Applications* **98** (1979), 1–26.

This is another aspect of dynamical systems I could get interested in. I’m always interested in the ‘arrow of time’, and there seems to be some nice math here.

]]>