WHAT IS a Strange Attractor?

1 A Strange Attractor? David Ruelle764 NOTICESOFTHEAMSVOLUME53, NUMBER7 Your computer will readily implement the map fsending the point (u,v) R2to the point(v+1 au2,bu) R2, where a= b= your computer to plot the points xn=fn(0,0),and you will find that they accumulate, for n ,on a convoluted fractal set Aknown as the H nonattractor. This set is prototypical of what one wantsto call a Strange attractor . Such objects often arisewhen a diffeomorphism fstretches and folds anopen set Uand maps the closure f Uinside U(thisis a typical situation, not a necessary or a sufficientcondition). The Strange attractor Ais visualizedwhen a computer plots the points xn=fnx0withalmost any initial value x0in U.

2 The figures showa two-dimensional example corresponding to theH non attractor , and a three-dimensional examplecorresponding to Smale s solenoid. It turns out thata small change in the values of aand bcan destroythe H non attractor , but a small change of fdoesnot destroy the give now a definition of uniformly hyperbolicattractors, or Axiom A attractors. This will cover thecase of the solenoid but not the H non attractor .(One can also define nonattracting hyperbolic sets,called basic sets, but they will not concern us.) Letfbe a diffeomorphism of the compact manifold M,and put some Riemann metric on M. Let Abe acompact invariant subset of M.

3 We say that Ais ahyperbolicset if one can continuously choose ateach point x Aa contracting (or stable) subspaceEsxand an expanding (or unstable) subspace EuxofTxMso that (Tf)Es,u=Es,uand TM=Es Eu. Wethus assume that||(Tf)X||<||X||if 0 =X Es||(Tf) 1X||<||X||if 0 =X ask that Abe attracting, , that Ahave an openneighborhood Usuch that t 0ftU=A(one can then arrange that f U U). Smale s AxiomA also requires that f-periodic orbits be dense inAand that Acontain a dense f-orbit (topologicaltransitivity).An Axiom A attractor is either a finite attract-ing periodic orbit, or it is an infinite set, and thedimension of the expanding subspaces Euxis > the latter case, Ais a strangeAxiom A attrac-tor: there are points yclose to xin Asuch that thedistance between fnxand fnygrows exponentiallywith nuntil this distance becomes of the order ofthe diameter of A.

4 The exponential growth ofdist(fnx,fny)expresses chaos, or sensitivity to ini-tial condition: if there is any imprecision on x, thepredictability of fnxis lost for large n. Here, fstretches the set Uand necessarily also folds this set to put f Uback in U, in agreement with thenotion of a Strange attractor . Careful studies haveshown that various systems that occur in natureare chaotic: their time evolution is described by low-dimensional dynamics with sensitivity to initialcondition, for which Strange Axiom A attractorsoffer an excellent mathematical model. In partic-ular, a proposal by Floris Takens and myself thathydrodynamic turbulence is chaotic in this sensewas eventually vindicated by experiment.

5 It is onthis occasion that the name Strange attractor seems to have been theory of uniformly hyperbolic attractors,initiated by Dmitrii Anosov and Stephen Smale,shows that one can define stable and unstablemanifolds VSx,Vux M: these are nonlinear ver-sions of the spaces Esx,Eux TxM, and they give aglobal meaning to contracting and expanding di-rections. One proves structural stability: if fisclose to f, then fhas an attractor Aclose to A, sothat f| Ais topologically conjugate to f|A. Using? what Ruelle is professor emeritus at the Institut des Hautes tudes Scientifiques, Bures-sur-Yvette, France, and Distin-guished Visiting Professor in the mathematics departmentat Rutgers University.

6 His email address is density of periodic orbits,Smale proved local product struc-ture: Ais locally the product of aset in the contracting direction anda set in the expanding direction.(The set in the expanding direc-tion is a manifold, while in the con-tracting direction it is usually afractal, for instance a Cantor set.)Yakov Sinai obtained a global con-sequence of the product structurecalled symbolic dynamics(an im-proved later treatment was givenby Rufus Bowen). Here is the result:up to well-controlled ambiguities,one can associate with each pointx Aan infinite sequence of sym-bols (.., 1, 0, 1,..). The iinthe sequence are taken from a finite set Fand onlycertain successive pairs ( i, i+1)are allowed, butno other condition is imposed.

7 Furthermore, re-placing xby fxreplaces (.., 1, 0, 1,..)by theshifted sequence (.., 0, 1, 2,..). Symbolic dynamics thus replaces geometry (the diffeomor-phism f) by algebra (the shift on sequences of sym-bols). For example, using symbolic dynamics, onecan count periodic points in A( , give a formulafor cardfn|A). One can also study f-invariant mea-sures on A, in particular so-called Gibbsmeasures:they correspond to thermal equilibrium states for an interacting one-dimensional spin systemasstudied in statistical mechanics when the sequence(.., 1, 0, 1,..)is interpreted as an infiniteconfiguration of spins iin one Lebesgue-almost-all points xin the neigh-borhood Uof A, the time averages along the for-ward orbit (x,fx.)

8 ,fnx,..)tend to a probabilitymeasure SRBon A, and this measure does not de-pend on x. This so-called SRB-measure is a Gibbsmeasure, introduced by Yakov Sinai, myself, andRufus Bowen. We have a detailed understanding ofSRB measures and how they depend on now about non-Axiom A attractors? Todiscuss them, we must introduce the wonderfulidea by Yakov Pesin that the geometric conditionof uniform hyperbolicity for dynamical systems canbe replaced by an almost-everywhere analysis withrespect to any given ergodic measure . Pesin the-ory allows one to define stable and unstable man-ifolds Vs,uxfor -almost-all x. Furthermore, thosemeasures that can be called SRB measures havebeen characterized by Fran ois Ledrappier, Jean-Marie Strelcyn, and Lai-Sang Young.

9 General SRBmeasures are a beautiful measure-theoretic ver-sion of Axiom A attractors, without the uniform hy-perbolicity assumption; they again describe timeaverages for a set of positive Lebesgue measure inthe manifold M. The support of a general SRB mea-sure is typically a fractal object that deserves to becalled a Strange attractor and is in fact what yousee when a computer plots for you, say, the H nonattractor. We may say that, to go beyond hyper-bolicity, we have replaced the geometric conceptof Strange attractor by the ergodic concept of great generality of Pesin theory comes at aprice: it is hard to know how things change whenthe diffeomorphism fis replaced by a nearby dif-feomorphism f.

10 Here we reach the current fron-tier in the theory of smooth dynamical systems. Cvitanovic has proposed a fascinating descriptionof the pruning front associated with changes inthe H non attractor . A theory of H non-like dif-feomorphisms has been developed (by Benedicks-Carleson, Viana, Young, and others) where theSRB measure is analyzed, not for all perturbations fof f, but for a set of positive measure of such per-turbations in some parameter space. Jacob Palis has proposed a beautiful set of conjectures to theeffect that for most diffeomorphisms f(in somesense), and most initial points x M(in the senseof Lebesgue measure), time averages correspondto a finite number of SRB measures (or attractorsif you like).