Resumen:
In the talk, I will show that for dynamical system from some open set, the
Milnor attractor has an unexpected geometrical structure.
The
likely limit set (also called Milnor attractor) of a map \(f:X\to X\) acting on a
metric space X with a measure is the minimal closed set \(A_M\) such that
\(f^n(x)\to A_M\) as \(n\to\infty\) for almost every point \(x\in X\).
Until 2010, for all known
typical dynamical systems, the Milnor attractor
was one of the following:
* a smooth submanifold (or the whole manifold);
* locally homeomorphic to a product of a manifold by a Cantor set;
* a Cantor book.
During my PhD studies, I constructed a non-empty open subset of \(\mathrm{Diff}(\mathbb T^3)\) such that each map from this subset has following unexpected properties:
* (expected) it is partially hyperbolic, and all leaves of the central
foliation are circles;
* its Milnor attractor intersects most of the central leaves in a single point
(the plot part of \(A_M\));
* its Milnor attractor intersects an uncountable set of central leaves on a segment (a bone);
* all bones are included by the closure of the plot part.
In other words, we have a function defined on a large (full measure) subset of
the torus that behaves similar to \(\sin(1/x)\) near uncountably many points. Our
Milnor attractor is the closure of the plot of this function. Yu. Ilyashenko
proposed to call attractors of this type
bony.
I shall talk about two generalizations of this construction to the higher
dimension case, one due to Ilyashenko, other is mine.
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