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## 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.