Yu. Ilyashenko and A. Negut have discovered the following dynamical phenomenon. Recall that the statistical attractor is the smallest closed set such that almost every orbit spends almost all its time arbitrarily close to . They say that an open set is -invisible if almost every orbit spends a fraction of its time less than in it.
-invisibility for very small may occur for trivial reasons: if 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 , they obtain a ball of radius set among a class of skew-products over the doubling map of the circle such that:
- the Lipschitz constant is independent of ;
- the map is structurally stable;
- the attractor covers ;
- the whole space above is -invisible
Ilyashenko and Volk have just published new examples with -invisibility.