Posts Tagged ‘attractors’

We consider a diffeomorphism f:M\to M where M is a compact manifold.

A topological attractor is a compact subset \Lambda\subset M which is (i) invariant: f(\Lambda)=\Lambda; (ii) chain recurrent: for any x,y\in\Lambda, any \epsilon>0, there exists a finite sequence x_0=x,x_1,\dots,x_N=y\in M such that d(f(x_i),x_{i+1})<\epsilon; and (iii) whose basin, \{x\in M:\lim_{n\to\infty} d(f^nx,\Lambda)=0\},  is a neighborhood of \Lambda. This last property can be stated as: it admits a neighborhood U such that \Lambda=\bigcap_{n\geq0} f^n(U).

One might be tempted to think that any (or almost every) diffeomorphism admits at least one attractor. This is the case on surfaces (according to Pujals-Sambarino) and in higher dimension far from homoclinic tangencies (according to Crovisier-Pujals).  However this fails in full generality.

More precisely, in a 2009 preprint, Bonatti, Li and Yang have built, on any compact manifold of dimension at least 3,  a locally C^1-generic diffeomorphism with no topological attractor (i.e., this property holds for all diffeomorphism in a subset containing a dense intersection of countably many open subsets of some fixed open subset of Diff^1(M)).

Rafael Potrie has explained to our work group (on June 21)  his construction of such an example (see the preprint here). His example is derived from Anosov. This  allows a precise control of its dynamics:

  • there is a unique minimal Milnor attractor. More precisely, it is a compact invariant set which is chain recurrent, whose basin has full Lebesgue measure and such that any invariant proper compact subset has a basin of zero Lebesgue measure.
  • each chain recurrent class besides the Milnor attractor is contained in a cycle of center-stable-leaves.

He carefully explained his construction. He starts with a hyperbolic automorphism of the 3-torus with a simple expanding eigenvalue and a pair of conjugate, non-real, contracting eigenvalues. As in Bonatti-Viana’s paper, one modifies it in a ball of some small radius \delta>0 around a fixed point q to get:

  • q becomes a saddle point with one wealky contracting eigenvalues and two expanding ones such that the area of any 2-plane is expanding. Also, its local stable curve has length larger than \delta;
  • there are thin unstable, resp., center-stable cones, invariant under f, resp., f^{-1}, and close to the unstable, resp. stable, spaces of the linear map.

According to Bonatti-Viana, there is a constant L<\infty such that, any center-stable disk of radius bigger than 2\delta and any unstable curve of length at least L intersect.

Recall that a quasi-attractor is a compact subset \Lambda satisfying (i), (ii) and the following weakening version of (iii): it admits a decreasing sequence of neighborhoods U_n such  that \Lambda = \bigcap_{n\geq0} U_n and \overline{f(U_n)} \subset  U_n. An application of Zorn’s Lemma shows that all homeomorphisms of compact sets have quasi-attractors.

Potrie shows that his diffeomorphism has a unique quasi-attractor by observing that any small piece of unstable curve contained in any U_n is eventually mapped into a long curve that must meet the large stable disk of a fixed point r away from p,q. It follows that any quasi-attractor contains r. Hence the uniqueness.

To prove that the quasi-attractor is not an attractor, Potrie observes first that it must contains q and that, generically by a result of Bonatti-Crovisier, it must coincide with the homoclinic class of q. The choice of the eigenvalues of the automorphism implies that the center-stable space of this class has no finer dominated splitting. T_xf|E^{cs}  contracts areas. By Bonatti-Diaz-Pujals, this generically implies that the class is accumulated by sources.

The proof that the Milnor attractor is minimal is done in two steps. For C^2 systems, it follows from the existence of a Sinai-Ruelle-Bowen measure.  For C^1 systems, it follows from the fact that the existence of a minimal Milnor attractor supported by the closure of the unstable set of r is a G_\delta, which is dense by the C^2 result.

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Yu. Ilyashenko and A. Negut have discovered the following dynamical phenomenon. Recall that the statistical attractor is the smallest closed set A such that almost every orbit spends almost all its time arbitrarily close to A. They say that an open set U is \epsilon-invisible if almost every orbit spends a fraction of its time less than \epsilon in it.

\epsilon-invisibility for very small \epsilon may occur for trivial reasons: if U is has a small intersection with the attractor, if the transformation (or its inverse) has very large Lipschitz constant, if it is close to a structurally unstable dynamics.

Ilyashenko with Negut have built non-trivial examples of such invisible subsets. For every integer parameter n\geq100, they obtain a ball of radius 1/n^2 set among a class of skew-products over the doubling map of the circle \mathbb Z/\mathbb R such that:

  • the Lipschitz constant is independent of n;
  • the map is structurally stable;
  • the attractor covers [1/n,1-1/n];
  • the whole space above ]0,1/4[ is 2^{-n}-invisible

Ilyashenko and Volk have just published new examples with 2^{-n^k}-invisibility.

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