arXiv subjects

Ivan Shilin

Publications and source records attributed to Ivan Shilin.

9 recordsLinked to original sources

arXiv2026

Generic families of circle diffeomorphisms have many coexisting periodic orbits

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 the diffeomorphisms have arbitrarily large finite numbers of periodic orbits. This phenomenon implies that families where irrational rotation numbers appear are not weakly structurally stable. Moreover, we prove that any locally residual set of one-parameter families with nonconstant rotation number yields a continuum of weak equivalence classes of families.

math.DS↗

arXiv2026

Diophantine "Tears of the Heart"

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, in a special one-parameter subfamily where the "heart" is preserved and the "tear" loop if broken, at least four invariants of weak topological classification appear. In this paper, we demonstrate that the metrical perspective yields a different result: for Lebesgue almost all values of the coefficients related to the original vector field, the special one-parameter family generates only two such invariants.

math.DS↗

arXiv2025

New Numerical Invariants of an Unfolding of a Polycycle "Tears of the Heart"

In this paper new numerical invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the pairs of the unperturbed arithmetic progressions we give a complete topological classification.

math.DS↗

arXiv2023

Attractors with non-invariant interior

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 contrast to the properties of diffeomorphisms, where the interior must be invariant.

math.DS↗

arXiv2022

Attractors with Non-Invariant Interior and Pinheiro's Theorem A

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 this research by Pinhero's Theorem A from his recent preprint, and in Section 3 we give a simple and straightforward proof of this result.

math.DS↗

arXiv2021

Attractors of direct products

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows $φ$ on $S^2$ for which the attractor of the Cartesian square of $φ$ is smaller than the Cartesian square of the attractor of $φ$. In the example for the minimal attractors, the flow $φ$ also has a global physical measure such that its square does not coincide with the global physical measure of the square of $φ$.

math.DS↗

arXiv2019

Sparkling saddle loops of vector fields on surfaces

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 bifurcations can differ drastically from the analogous bifurcations on the sphere. The reason is that, on a surface, a free separatrix of a hyperbolic saddle may wind toward the separatrix loop of the same saddle. When this loop is broken, sparkling saddle loops emerge. In the orientable case, the parameter values corresponding to these loops form the endpoints of the gaps in a Cantor set contained within the bifurcation diagram. Due to the presence of a Cantor set, there is a countable set of topologically non-equivalent germs of bifurcation diagrams even in generic one-parameter families, in contrast to bifurcations on the sphere.

math.DS↗

arXiv2017

On the attractors of step skew products over the Bernoulli shift

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 skew product the statistical and the Milnor attractor coincide and are Lyapunov stable. For this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. For the case of the fiber being a segment we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the skew product to the fiber.

math.DS↗

arXiv2016

Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors

We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E. R. Pujals that, for a $C^1$ topologically generic diffeomorphism of a closed manifold, either any homoclinic class admits some dominated splitting, or this diffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism has an unstable Milnor attractor. The same results hold for statistical and minimal attractors.

math.DS↗