VII. Quadratic maps of the interval and the Hénon attractor

The first mathematical studies of what we would now call chaotic systems were for hyperbolic systems: the Axiom A systems of Smale are hyperbolic and the Lorenz attractor is also basically hyperbolic. The ergodic theory of dynamical systems developed following the work of Pesin provides a vast generalization: what corresponds to hyperbolicity in Pesin's theory is the fact that an ergodic measure has no characteristic (or Lyapunov) exponents equal to zero. Pesin's theory is, however, basically non-constructive.

What can we then do about specific systems? Take the map x

ax(1-x) of the interval [0,1] to itself (with 0<a£4). This map (for a close to 4) is not hyperbolic, and it turns out that its study poses very difficult problems. When is such a system chaotic? The first breakthrough in this area is due to Jakobson [24] who proved the following result: there is a set of positive Lebesgue measure of values of the parameter a such that the system x®ax(1-x) has an ergodic measure absolutely continuous with respect to the Lebesgue measure on [0,1]. Jakobson's proof has been improved by Benedicks and Carleson [25] and we now know that the ergodic measure in question can be taken to have positive characteristic exponent, so that the system is indeed chaotic.

To study the map x®ax(1-x) for a close to 4 the basic difficulty is this: while the map is mostly stretching, it is in fact contracting for x close to 1/2: we do not have hyperbolicity. Some "bad" intervals of values of a are such that the orbits pass repeatedly close to 1/2, and therefore this destroys stretching. It is however possible to show that there are values of a outside of all "bad" intervals, and that the set of these values has Lebesgue measure > 0. This requires careful "Swiss watchmaker" type work, and yields the results mentioned above.

Benedicks and Carleson [26] then went on to a more difficult problem, that of some quadratic maps of the plane R² to itself called Hénon mappings. Essentially, these are the quadratic maps of R² with constant Jacobian, and Hénon has shown numerically that some of them have strange attractors. This is what Benedicks and Carleson were able to prove mathematically. Their work was later improved and expanded by Mora and Viana [27].

To put things in perspective, let me stress that the mathematical work referred to in this section is very hard, and just makes contact with the simplest examples of chaos obtained from equations that one would naturally write down. This shows how little intellectual control we still have on chaos: the problems are really difficult, and incite us to be modest.