2018-11-18  10:59 hrs.

### Bony attractors in higher dimension

#### Yury KudryashovHigher School of Economics

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.

#### Palapa Guillermo Torres -- Jueves 8 de mayo de 2014, 16:00 horas

Unidad Cuernavaca del Instituto de Matemáticas UNAM