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## Rafael Potrie: Locally generic Diffeomorphisms With No Attractor

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.

## Invisible Attractors

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.