Popular presentations of Carleson's results

The Hénon map

The most recent work that the committee singles out in its decision to give the Abel Prize to Lennart Carleson originates in the period 1985-1991 and culminates in Carleson and Benedick's dissertation from 1991, where they prove that the Hénon map has a strange attractor. This work lies within the field of dynamic systems. In order to get an insight into what it is about, we shall go back to 1960, to MIT in the USA, where the meteorologist Edward Lorenz was involved in creating good weather models, as meteorologists tend to do. Lorenz had what by today's standards we would call an extremely primitive computer to help him perform the enormous number of calculations that were required in order to predict the weather. Roughly speaking, modern weather forecasting entails considering the physical laws that apply and the initial conditions we have for wind, humidity, pressure etc. right now. Using this description, we calculate the magnitude of these same parameters a short time interval later, then another time interval after that, another after that, etc. until we end up with a forecast of tomorrow's weather. Lorenz had to simplify the whole model down to three parameters, he gave each of these three their value and then he began to “crank the machine”, i.e. make repeated calculations.

The story goes that Lorenz tried one day to continue a run he had started the day before. He began about half way as far as he had come, entered the relevant numbers and started the machine. At first, everything agreed with his observations from the day before, but suddenly the values began to deviate from the previous day's numbers, at first only a little, but then the deviation accelerated rapidly and before he knew it, the model had predicted something completely different from what it had done the day before. How could this happen? The equations were the same, the starting point was the same, the computer was the same, yet the response was different?

The explanation was that they were not the same values. Lorenz rounded off the fourth decimal place when he started on the second day. This meant that the initial conditions were slightly different, but could a difference of one 10,000th of a per cent cause such a catastrophe? We usually take it for granted that a small difference in input gives a small difference in output, but this was not the case here. The reason was that the process was based on repetitions where the previous result became the next premise. A small deviation which is slightly magnified at each step, will eventually lead us into the unknown after many steps. Lorenz had discovered the phenomenon that has come to be known as the butterfly effect in meteorology, namely that a single flap of the wings of a butterfly in Beijing in March can cause the August hurricanes in the Atlantic Ocean to follow a completely different course!

We shall set aside all of the meteorological and other physical consequences of Lorenz’s discovery and focus on its mathematical content. By making use of powerful computers, it was not difficult to create illustrations of the Lorenz system; but this did not give us any particular insight into the mathematical structure, nor did it look as if anyone would be able to provide that insight.

In 1976, the astronomer Michel Hénon presented a simplified version of Lorenz's system. Hénon's discrete dynamic system had two important ingredients. It was much easier to calculate with than the Lorenz system, and like the Lorenz system it had a strange attractor. The Hénon system is described with a map T of a plane into itself expressed by the rule

(it is okay to use other coefficients instead of 1.4 and 0.3, but these are the ones that are usually used in examples). According to this rule, the point (0, 0) is plotted at the point (1, 0), the point (1, 0) is plotted at the point (-0.4, 0.3), which in turn is plotted at the point (1.076, -0.12), etc. In this case, we end up inside the attractor, the curve that is illustrated in the figure below. The same is true if we begin with the point (0, 0.2918), but not if we begin with (0, 0.2919). In the latter case, we will rapidly disappear far out toward infinity. This is easy to program in a spreadsheet. Enter the following in the four squares at the upper left of the spreadsheet:

Now copy the fields A2 and B2 below in columns A and B as far as you want to go, for 10,000 rows if you like. If we now label all of the fields A1:B10000 and print the graphic symbol at the top of the spreadsheet, choose XY (scatter) and then the alternative without any connecting lines, then we get a map of the Hénon attractor, roughly like the figure below.

The points in this set are distributed in an apparently unsystematic way: one here, one there. It is only when we approach several thousand points that we begin to see the major contours. It turns out that what we think we see is not the whole truth. If we enlarge the slightly thick lines in the attractor, new details steadily emerge: the lines are not single strings, but multiple strings. If we continue to enlarge, we see that the same thing repeats itself, the single strings always split up into even smaller single strings. We see that the attractor has fractal properties.

This is where the really difficult question arises. If we make 10,000 calculations in this way, or if we make 10,000,000 calculations, we get about the same picture – more calculations will only reveal more of the fractal structure. But how do we know that we will not suddenly fly off into infinity, as we did after we had made about 35 iterations when the starting point was (0, 0.291807922563607)? Is this really an attractor?

Carleson and Benedicks presented a formal proof that the world is exactly as we think it is, that a strange attractor does exist. Nothing unforeseen is going to happen even if we make quadrillions of iterations. Once we are inside this strange set, we will remain there. The problem may seem rather contrived and narrow to a non-mathematician, but this is mathematics, not meteorology or physics, and therefore it is imperative that we know and not just believe.

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