# Attractor

Post-publication activity

Curator: John W. Milnor

Roughly speaking, an attracting set for a dynamical system is a closed subset $$A$$ of its phase space such that for "many" choices of initial point the system will evolve towards $$A\ .$$

Figure 1: The unit circle as an attractor for a flow in the plane. (In this example, distant points get pulled in very quickly, but points near the origin are slow to converge.)

(Words shown in red refer to future links which are not yet operational.) The word attractor will be reserved for an attracting set which satisfies some supplementary condition, so that it cannot be split into smaller pieces. In the case of an iterated map, with discrete time steps, the simplest attractors are attracting fixed points. Similarly, for solutions of an autonomous differential equation, with continuous time, the simplest examples are attracting equilibrium points. In both cases, the next simplest examples are attracting periodic orbits. (See Figure 1 for the continuous time case, and Figure 5 for discrete time.) However, much more exotic attractors exist and are important ( Figure 4).

Care is needed since the literature contains many variations on the precise definitions. (For example, many authors make no distinction between attractors and attracting sets.) This exposition will emphasize first the classical definitions, and then the more recent measure theoretic definitions. In either case, the union of all orbits which converge towards $$A$$ is called the basin of attraction $$B(A)\ .$$ The exposition concludes with a rather different statistical definition.

As an example, Figure 1 illustrates the flow in the complex plane generated by the differential equation $$\dot z =(1+2i-|z|^2)z\ .$$ The initial image shows the $$4\times 4$$ square centered at the origin, filled with a cloud of dots, and the remaining pictures show succesive positions of these dots under the flow. The trajectory (= orbit) of every point except the origin converges to the unit circle, which is an attracting periodic orbit.

(Caution. The word "attractor", as used in dynamical systems, has nothing at all to do with its use in gravitational theory (see Great Attractor). Attractors in dynamical systems theory simply provide a way of describing the asymptotic behavior of typical orbits. In particular, there is no associated attractive force.)

## Classical Attracting Sets: Three equivalent definitions

To simplify the discussion, this article will concentrate on discrete-time dynamical systems, consisting of a locally compact metric space $$X$$ called the phase space, together with a function $$f$$ from $$X$$ to itself which describes the evolution of the system in one time step. However, there are completely analogous definitions for systems with continuous time, which are usually defined by autonomous differential equations. (For attractors of ordinary differential equations, see for example Hurley [1982]; and for partial differential equation attractors see Temam [1997].)

The easiest definition to work with is the following.

### Trapped Attacting Sets

Let $$T\subset X$$ be a compact set such that $$f(T)$$ is contained in the interior of $$T\ .$$ Then the intersection $$A$$ of the nested sequence of sets

$$T~\supset~f(T)~\supset~f^{\circ 2}(T)~\supset~\cdots$$

will be called a trapped attractor, with $$T$$ as trapping neighborhood. (Here $$f^{\circ n}$$ stands for the $$n$$-fold iterate of $$f\ .$$) This intersection is always invariant, $$f(A)=A\ ,$$ and its attracting basin $$B(A)$$ is always an open set containing $$T\,.$$

Figure 2: The grey annulus is a trapping region $$T\ ,$$ with image $$f(T)$$ shown in red.
Figure 3: The corresponding attractor is shown in red.

A simple example of a trapped attracting set is shown in Figure 3. Here the phase space $$X$$ is the plane with the origin removed, the map $$f$$ is given in polar coordinates by the formula $$(r,\theta)\mapsto(1.5+.25r+.5\cos\theta, 2\theta)\ ,$$ the trapping region $$T$$ is the annulus $$1\le r\le 3\ ,$$ and the basin $$B(A)$$ is all of $$X\ .$$

Such trapped attracting sets have a very convenient robustness property:

If $$g$$ is another map from $$X$$ to itself which is uniformly close to $$f\ ,$$ then there is a trapped attracting set $$A'=\bigcap g^{\circ n}(T)$$ for $$g\ ,$$ with $$A'$$ contained in a small neighborhood of $$A\ .$$

### Isolating Neighborhoods

By a neighborhood of a subset $$A\subset X$$ we mean a subset of $$X$$ which contains $$A$$ in its interior. By a forward isolating neighborhood of a compact invariant set $$A$$ we mean a neighborhood $$N$$ of $$A$$ such that $$A$$ is equal to the intersection of the forward images $$f^{\circ n}(N)\ .$$ (Compare Smale [1967], or Conley [1978].) The existence of such a forward isolating neighborhood seems to be a much weaker requirement than existence of a trapping neighborhood; but in fact it is completely equivalent:

A compact invariant set has a trapping neighborhood if and only if it has an isolating neighborhood.

In fact every isolating neighborhood contains a trapping neighborhood (and conversely any trapping neighborhood is itself an isolating neighborhood). This can be proved by first showing that any $$A$$ with an isolating neighborhood has arbitrarily small neighborhoods $$N$$ with $$f(N)\subset N\ .$$ (A proof is given for example in Milnor [1985b].) Given such an $$N$$ which is compact and attracted to $$A\ ,$$ let

$$\delta(x)~=~\sum_0^\infty 2^n \,{\rm dist}\Big(f^{\circ n}(x), N\Big)\qquad$$   for   $$x\in B(A)\,.$$

Then $$\delta(f(x))\le \delta(x)/2\,.$$ The set $$\{x\,;\,\delta(x)\le\epsilon\}$$ is compact for small $$\epsilon\ ,$$ and provides the required trapping neighborhood.

### Lyapunov Stablility

In the original definition, due to Auslander, Bhatia, and Seibert [1964], a compact $$f$$-invariant subset $$A=f(A)\subset X$$ is called a Lyapunov stable attracting set if it has an open basin of attraction, and if the following condition is satisfied:

• Lyapunov Stability.  Every neighborhood $$U$$ of $$A$$ contains a smaller neighborhood $$V$$ such that every iterated forward image $$f^{\circ n}(V)$$ is contained in $$U\ .$$

Using the discussion above, it follows easily that:

The compact invariant set $$A$$ is a Lyapunov stable attracting set if and only if it has a trapping neighborhood, or if and only if it has a forward isolating neighborhood.

Thus the concept of "attracting set" can be defined in three different but completely equivalent ways.

Figure 4: This set, discovered by Yoshisuke Ueda in 1961, appears to be a trapped attractor. Here the $$(x,\,\dot x)$$ plane is mapped into itself by following the trajectory of the Duffing equation $$\ddot x+.05\,\dot x+x^3=7.5\,\cos(t)$$ for time $$0\le t\le 2\pi\ .$$

## Classical Attractors and Repellors.

The word attractor is usually reserved for an attracting set which contains a dense orbit. (This condition insures that it is not just the union of smaller attracting sets.)  As an example, the trapped attracting set shown in Figure 2 and Figure 3 is certainly an attractor in this sense.  In fact it is an example of what Ruelle and Takens [1971] have called a strange attractor (see also Ruelle [2006]). That is, the dynamics is chaotic, depending sensitively on initial conditions. Like many strange attractors, this set $$A$$ is also an example of a fractal, with complicated structure near any point, under any scale of magnification. Perhaps the first such examples were the Lorenz attractors and the Hénon attractors, which have been widely studied, and the Ueda attractors ( Figure 4).

Just as nearby orbits converge towards an attractor, they diverge away from a repellor. By definition, a repelling set is a compact invariant set $$K$$ which possesses a backwards isolating neighborhood; that is a neighborhood $$U$$ such that $K~=~\bigcap_{n\ge 0} (f^{\circ n})^{-1}(U)\,.$ In other words, given any point $$x_0\in X\smallsetminus K\ ,$$ the orbit $$x_0\mapsto x_1\mapsto\cdots$$ must satisfy $$x_n\not\in U$$ for at least one value of $$n\ge 0\ .$$ This definition works both for invertible and non-invertible maps. A repelling set with a dense orbit will be called a repellor.

There is one curious difference between attractors and repellors. If two trapped attractors have a single point in common, then it is not hard to see that they must be identical to each other. But repellors can even be nested, one within the other. ( Figure 5.) This difference may be observed for non-invertible maps only.

Figure 5: The $$z$$-plane for the complex polynomial map $$f(z)=z^2-1\ .$$ The points zero and -1, marked by red dots, form an attracting period two orbit with attracting basin colored grey. If we compactify the plane by adjoining a point at infinity, then the fixed point at infinity is also an attractor, with basin shaded orange-to-green. The common boundary of these two basins, colored black, is a repellor called the Julia set. It contains infinitely many periodic orbits, and each one is also a repellor.

## Measure Attractors

It is often reasonable in dynamical systems theory to ignore any behavior which occurs only on a set of measure zero, since such behavior will never be observed in any real world application. This suggests the following. Suppose now that the phase space $$X$$ is a smooth manifold (for example an open subset of Euclidean space), so that there is a well defined distinction between sets of measure zero and sets of positive measure. If $$X$$ is not compact, then it will be convenient to compactify by adding a point at infinity.

Definition.  A compact set $$A\subset X$$ (or $$A\subset X\cup\infty$$ if $$X$$ is not compact) will be called a   measure attracting set   if:

• the basin of attraction $$B(A)\ ,$$ consisting of all points whose orbits converge towards $$A$$ has strictly positive measure; and furthermore
• for any closed proper subset $$A'\subset A\ ,$$ the set theoretic difference $$B(A)\smallsetminus B(A')$$ also has strictly positive measure. (This last condition is needed in order to guarantee that every part of $$A$$ plays an essential role. Note that, in this theory, the basin need not be an open set.)

(Compare Milnor [1985a]). Here an orbit $$\{x_n\}$$ is understood to converge to $$A$$ if the distance between $$x_n$$ and $$A$$ tends to zero as $$n\to\infty$$ (using a suitable metric on $$X\cup\infty$$ if $$X$$ is not compact).

One useful feature of this definition is that every such dynamical system has a unique largest measure attracting set $$\widehat A$$ called the global attracting set. This can also be described as the smallest closed set with the property that every orbit outside of a set of measure zero converges towards $$\widehat A\ .$$

A measure attracting set is called minimal if no proper closed subset has a basin of positive measure. Since two distinct minimal attracting sets must have disjoint basins of positive measure, there can be at most countably many of them. The word attractor will be reserved for a minimal attracting set which contains a dense orbit, and hence cannot be expressed as a union of smaller closed invariant sets. (However two distinct attractors, in this sense, may well intersect each other. See Alexander, Hunt, Kan and York [1996]; or Bonifant, Dabija, Milnor [2006].) In nice cases, the global attracting set will be a union of measure attractors, each with a well defined asymptotic measure. But this is not true in all cases: As a boring example, for the projection map $$(x,y)\mapsto(x,0)$$ in the plane, the global attracting set is the $$x$$-axis; but there are no attractors or minimal attracting sets at all. See Palis [2005] for a collection of challenging conjectures and a report about recent work concerning attractors for typical dynamical systems.

Figure 6: The global attractor for the complex exponential map consists of the points $$0,\,1,\,e,\,e^e,\,\ldots\,,$$ together with the point at infinity.

Here are three examples.

### Example: The complex exponential map

The map $$\exp:{\mathbb C}\to{\mathbb C}$$ can be defined by the formula $$\exp(x+iy)=e^x\big(\cos(y)+i\,\sin(y)\big)\ .$$  For every initial point $$z_0\in{\mathbb C}$$ outside of a set of measure zero, Lyubich [1987] and Rees [1986] have shown that the set of all accumulation points for the orbit $$z_0\mapsto z_1\mapsto z_2\mapsto\cdots$$ within $$\mathbb C\cup\infty$$ is just the orbit of zero, consisting of the points
$$0,\; 1,\;e,\;e^e,\;e^{e^e},\;\ldots~\in~{\mathbb C}\,,$$

together with the point at infinity. Thus the unique measure attractor is the set $$A=\{\infty,\,0,\,1,\,e,\,e^e,\,\ldots\}\,.$$

It is amusing to test this statement numerically. On a computer, an orbit of the complex exponential map with randomly chosen non-real starting point will usually land exactly at zero after a relatively small number of iterations, unless there is an overflow error first. Of course a true (infinite precision) orbit could never land exactly at zero. Furthermore, not all true orbits converge to the orbit of zero.  In fact, Misiurewicz [1981] has shown that there is an uncountable dense set of initial points $$z_0$$ such that the orbit of $$z_0$$ is everywhere dense in $$\mathbb C\ .$$ However, such dense orbits lie in a set of measure zero, and will never be seen experimentally.

Figure 7: Graph of the degree 4 Fibonacci map, with the critical orbit closure in red along the $$x$$-axis. This is a very thin Cantor set.

### Example: Fibonacci maps of the interval.

Bruin, Keller, Nowicki and van Strien [1996] have described similar wild behavior for certain maps of the interval. Consider polynomials of the form $$f(x)=c-x^{2n}$$ where $$1\le c^{2n-1}\le 2$$ so that $$f$$ maps the interval $$X=[f(c),\,c]$$ onto itself. Here the constant $$c$$ can be carefully chosen so that the critical orbit $$0=x_0\mapsto x_1\mapsto\cdots$$ returns closer and closer to zero after each Fibonacci number of iterations. ( Figure 7.) It then follows that $$f$$ has a dense orbit. However, the closure of the critical orbit is a Cantor set $$K\ .$$ If the degree $$2n$$ is large enough, they show that Lebesgue almost every orbit converges towards $$K\ ;$$ so that $$K$$ is the unique measure attractor.

### Example: Intermingled basins

The following example is due to Ittai Kan [1994]. As phase space take the cylinder $$S^1\times I\ ,$$ where $$S^1$$ is the circle with angular coordinate $$\theta$$ and $$I$$ is the unit interval. Let $f(\theta, y)~=~\big(2\theta,~y +\cos(\theta)y(1-y)/2\big)\,.$ Then there are two measure-theoretic attractors, namely the two boundary circles. Their union forms the global attracting set. However, the basins for the two attractors are intermingled in the sense that every nonempty open set intersects each basin in a set of strictly positive measure. (Compare Bonifant and Milnor [2006].)

Figure 8: The Kan example with two intermingled basins, colored red and blue. (Of course no computer picture can really distinguish between two such intermingled basins.)

### Remark

The term attractor is sometimes used for any set which contains a dense orbit and has an attracting basin of positive measure, omitting the second condition in the definition of measure attractor. However, this is a bad idea - some additional restriction is really needed. Otherwise, for the three examples described above, the entire phase space would qualify as an attractor.

## Statistical Attractors

The following discussion is based on the work of Ilyashenko, who has suggested several possible modifications of the basic definitions, emphasizing the behavior of "most" points of a typical orbit $$\{x_n\}$$ rather than considering only the limiting behavior as $$n\to\infty\ .$$ (See for example Ilyashenko [2005].) Recall that an orbit $$\{x_n\}$$ converges (in the usual sense) towards a compact set $$~A~$$ if the distance $${\bold d}(x_n,\,A)$$ tends to zero as $$n\to\infty\ .$$

Definition. The orbit $$~x_0\mapsto x_1\mapsto\cdots~$$ converges statistically towards $$A$$ if the time average of distances tends to zero,

$${1\over n}\Big({\bold d}(x_0,\,A)+\cdots+{\bold d}(x_{n-1},\,A)\Big) \to 0 \qquad {\rm as } \quad n \to\infty.$$

Thus occasional orbit points are allowed to wander away from $$A\ ,$$ as long as most of them converge. Here the distance function $${\bold d}(x,y)$$ should always be uniformly bounded, so that an occasional very distant orbit point will not affect the definition. The statistical global attracting set $$A^{\rm stat}$$ can be defined as the smallest closed set with the following property: For all $$x_0$$ outside of a set of measure zero, the orbit of $$x_0$$ converges statistically towards $$A^{\rm stat}\ .$$

In many cases, $$A^{\rm stat}$$ will coincide with the global attracting set $$\widehat A\ ,$$ as defined earlier. However, this is not always the case. As an example, for the complex exponential map, most orbits spend most of the time very far away from zero: Almost all orbits converge statistically towards the point at infinity. Thus the statistical global attracting set consists of the single point $$\infty\ .$$ For example, for almost every orbit $$\{z_n\}$$ it follows from the results of Rees or Lyubich, not only that there are infinitely many integers $$n_k$$ for which $$|z_{n_k}|<1/2\ ,$$ but also that the resulting sequence $$|z_{n_k}|$$ converges to zero as $$k\to\infty\ .$$ Yet the closer an orbit point gets to zero the more closely the orbit then shadows the orbit of zero, shooting out towards $$+\infty\ ;$$ and hence the longer it takes to get away from this diverging orbit and return once more to a neighborhood of zero. Thus the time differences $$n_{k+1}-n_k$$ tend to infinity, which implies that the origin is not in the statistical attracting set.

Hofbauer and Keller [1990] have described an example of a quadratic map of the interval with the following startling property: Almost all orbits converge statistically to a single repelling fixed point. Thus again the statistical global attracting set is a single point, although the global attracting set $$\widehat A$$ is much larger.