arXiv subjects

Stanislav Minkov

Publications and source records attributed to Stanislav Minkov.

5 recordsLinked to original sources

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↗