Uniformly hyperbolic attractors

• Dr. Todd Fisher, Department of Mathematics, Brigham Young University, Provo, UT

An attractor is a set \(\Lambda\) such that all points sufficiently close to \(\Lambda\) are forward asymptotic to \(\Lambda\ .\) A uniformly hyperbolic attractor is an attractor with a dense orbit and a hyperbolic structure. Uniformly hyperbolic attractors are remarkable in both their topological and statistical properties.

Background

A hyperbolic set for a diffeomorphism is a compact invariant set whose tangent space splits into uniformly expanding and contracting directions. For each point \(x\) in a hyperbolic set \(\Lambda\) there exist injectively immersed Euclidean spaces, that are as smooth as the diffeomorphism, called the stable and unstable manifolds. The stable manifold consists of all points that are positively asymptotic to \(x\) and the unstable manifold consists of all points that are negatively asymptotic to \(x\ .\) Furthermore, for \(x\in\Lambda\) the stable manifold at \(x\) is tangent to the contracting direction of the spitting at \(x\) and the unstable manifold at \(x\) is tangent to the expanding direction of the splitting at \(x\ .\)

A uniformly hyperbolic attractor, \(\Lambda\ ,\) is a hyperbolic set such that there is a point \(x\in\Lambda\) whose orbit is dense in \(\Lambda\) and a neighborhood \(U\) of \(\Lambda\) where \(f(\overline{U})\subset U\) and \(\Lambda=\bigcap_{k\geq 0}f^k(U)\ .\) Such a neighborhood \(U\) is called an attracting set. The set of all points that are positively asymptotic to a hyperbolic attractor is called the basin of attraction.

All uniformly hyperbolic attractors have certain basic properties.

• Periodic points are dense in the attractor.
• The unstable manifold of each point is contained in the attractor.
• The union of the stable manifolds for the orbit of a periodic point is dense in the basin of attraction for the hyperbolic attractor.

Examples

There are many standard examples of hyperbolic attractors including

• the Plykin attractor,
• Anosov diffeomorphisms,
• the solenoid (or Smale-Williams attractor), and
• the DA (derived from Anosov) - attractor.

As all but the last are described in other articles we give a brief description of the DA –Attractor. Let \(f_A:\mathbb{T}^2\rightarrow \mathbb{T}^2\) be the map on the torus induced by the linear map \(A=\begin{bmatrix}2 & 1\\ 1 & 1 \end{bmatrix}.\) Let \(p\)

Figure 1: Saddle fixed point

be a saddle fixed point of \(f_A\ .\) Using an appropriate bump function one can deform the map \(f_A\) in a sufficiently small neighborhood \(U\) of \(p\) such that

Figure 2: Deformed fixed point source

• the deformation is supported in \(U\ ,\)
• \(p\) is a source for the deformation, and
• the recurrent points for the deformed map consists of \(p\) and a uniformly hyperbolic attractor \(\Lambda\ .\)

The deformed map \(f\) is called the DA-diffeomorphism and the attractor is called the DA-attractor.

Properties

As stated in the beginning of this article the properties of uniformly hyperbolic attractors can be studied both for the topological and statistical properties.

In 1967, Smale constructed the DA – attractor as example of a structurally stable diffeomorphism that was neither Anosov or Morse-Smale. In the same paper the solenoid, previously known by topologists, was introduced as a uniformly hyperbolic attractor. In the same year Williams independently introduced the solenoid.

Williams analyzed the topological structure of uniformly hyperbolic attractors. Specifically, he studied the case where the topological dimension of the attractor was equal to the dimension of the unstable manifolds, called expanding attractors. Williams showed that locally an expanding attractor is the cross product of a Cantor set and the unstable manifold. He also showed that expanding attractors can be represented as inverse limits of a map on a branched manifold.

Plykin studied uniformly hyperbolic attractors on surfaces and constructed one-dimensional attractors on surfaces. He showed that the basin of attraction of a uniformly hyperbolic attractor on a surface must have at least 3 holes.

Another class of expanding attractors that have been studied extensively are the codimension-one attractors, those in which the stable manifolds are one dimensional. Plykin showed that codimension-one expanding attractors are generalized versions of DA-attractors. At present there is a complete classification of codimension-one expanding attractors. However, for general attractors on manifolds of dimension greater than or equal to three there are basic questions on the classification that are not well understood.

In the 1970’s Sinai began to look at the statistical behavior of Anosov diffeomorphisms. He noticed that although an Anosov diffeomorphism is deterministic it behaves like a stochastic system. His work was extended by Ruelle and Bowen to hyperbolic attractors. They showed that if \(f\) is a \(C^2\) diffeomorphism and \(\Lambda\) is a uniformly hyperbolic attractor for \(f\ ,\) then there exists a unique invariant probability measure \(\mu\) such that for every continuous function \(\varphi\) and Lebesgue almost every \(x\) in the basin of attraction for \(\Lambda\) the following is satisfied\[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^j(x))=\int\varphi d\mu.\] Such a measure is called an Sinai-Ruelle-Bowen (SRB) measure or physical measure. Sinai, Kifer, and Young were able to represent the SRB measure as the zero-noise limit of a stochastic process.

Additionally, Ruelle, Sinai, and Bowen were able to prove the central limit theorem and that the decay of correlations occurs exponentially fast for hyperbolic attractors of \(C^2\) diffeomorphisms.

References

• R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, vol. 470, Lect. Notes in Math., Springer Verlag, 1975
• V.Z. Grines, E. V. Zhuzhoma, Expanding attractors, Regul. Chaotic Dyn., 11 (2006), no. 2, 225-246
• Yu. Kifer, Random perturbations of dynamical systems, Birkhäuser, 1988
• Ya. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-69
• S. Smale, Differentiable Dynamical Systems. Bulletin of the American Mathematical Society (N.S.) 73 (1967), 747-817

Internal references

• John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.

• Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.

• Nikolai Chernov (2008) Decay of correlations. Scholarpedia, 3(4):4862.

• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

See also

Hyperbolic dynamics, Non-uniformly hyperbolic attractors

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2008-04-22 02:44:11 GMT