8 papers
Generic families of circle diffeomorphisms have many coexisting periodic orbits
Ivan Shilin
We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which t…
Diophantine "Tears of the Heart"
Yulij Ilyashenko, Stanislav Minkov, Ivan Shilin
Recent studies of topologically generic unfoldings of vector fields featuring a "tears of the heart" polycycle with one internal and one external winding separatrix have shown that…
Attractors with non-invariant interior
Stanislav Minkov, Alexey Okunev, Ivan Shilin
We construct an open set of endomorphisms of an arbitrary two-dimensional manifold which have attractors and non-wandering sets with non-invariant interior. This is a notable contr…
Attractors with Non-Invariant Interior and Pinheiro's Theorem A
Stanislav Minkov, Alexey Okunev, Ivan Shilin
This is a provisional version of an article, intended to be devoted to properties of attractor's intertior for smooth maps (not diffeomorphisms). We were originally motivated for t…
Attractors of direct products
Stanislav Minkov, Ivan Shilin
For Milnor, statistical, and minimal attractors, we construct examples of smooth flows on for which the attractor of the Cartesian square of is smaller than the Carte…
Sparkling saddle loops of vector fields on surfaces
Ivan Shilin
We study bifurcations of vector fields on 2-manifolds with handles in generic one-parameter families unfolding vector fields with a separatrix loop of a hyperbolic saddle. These bi…
On the attractors of step skew products over the Bernoulli shift
Alexey Okunev, Ivan Shilin
The statistical and Milnor attractors of step skew products over the Bernoulli shift are studied. For the case of the fiber a circle we prove that for a topologically generic step…
Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors
Ivan Shilin
We prove that for every smooth compact manifold and any , whenever there is an open domain in exhibiting a persistent homoclinic tangency related…