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. 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.

2 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. 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.

3 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. 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.

4 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. 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.)

5 The iinthe sequence are taken from a finite set Fand onlycertain successive pairs ( i, i+1)are allowed, butno other condition is imposed. 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,..,fnx,..)tend to a probabilitymeasure SRBon A, and this measure does not de-pend on x.

6 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. 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.

7 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. 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).

8 But it is also conceivable that for a largeset of diffeomorphisms fand of points x, the timeaverages are not even defined!The beautiful pictures above are by Bill Cassel-man and David Reading[1] ECKMANNand D. RUELLE, Ergodic theory of chaos andstrange attractors, Rev. Mod. (1985), 617 656.[2] YOUNG, what are SRB measures, and which dy-namical systems have them?, J. Statist. (2002), 733 754.[3] C. BONATTI, L. DIAZ, and M. VIANA, Dynamics Beyond Uni-form Hyperbolicity: A Global Geometric and Proba-bilistic Approach, Springer, Berlin, H non map fsends thequadilateral Uinside sets fU, f2U, f3 Uareplotted, and f3 Ualreadylooks a lot like the H nonattractor t s solenoid map fsends the torus Uinsideitself. Shown are U, fU, f2U, and you canimagine the solenoid t 0fnU.