arXiv subjects

Constantine Medynets

Publications and source records attributed to Constantine Medynets.

3 recordsLinked to original sources

arXiv2026

Characters and $II_1$-Factor Representations of Full Groups of Cantor Minimal Systems

Let $(X,T)$ be a Cantor minimal system, and let $\Gamma$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable (extreme) characters and the associated $\textrm{II}_1$-factor representations for the group $\Gamma$ and its commutator subgroup $\Gamma'$. In particular, we prove that: (1) for every nontrivial indecomposable character $\chi$ of $\Gamma'$, there exists a finite collection (with repetitions allowed) $\{\mu_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $\chi(\gamma) = \prod_{i\in I} \mu_i(Fix(\gamma))$, for every $\gamma \in \Gamma'$, where $Fix(\gamma) = \{x\in X : \gamma x = x\}$; and (2) each indecomposable character of $\Gamma$ is the product of an indecomposable character of the form $\prod_{i\in I} \mu_i(Fix(\gamma))$ and a homomorphism from $\Gamma$ into the unit circle. As a consequence, we show that any finite-type unitary representation of $\Gamma'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $\Gamma'$. We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs.

math.GR↗

arXiv2022

On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems

The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In the paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model $\ddot r_k = -p_k(|\dot r_k|)\dot r_k - \sum_m a_{k,m}r_m$, where $r_k\in \mathbb R^d$, $k=1,\ldots,n$, and $A = \{a_{k,m}\}$ is a symmetric positive-semidefinie matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than $1/z$, that is, $p_k(z)z\to\infty $ as $z\to\infty$. We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when $\ker (A)$ is trivial, the positions of the particles are also ultimately bounded. For systems with $\ker (A)\neq \{0\}$, we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto $\ker (A)$. We also include the proof of the ultimate boundedness of velocities and accelerations for systems with bounded coupling, including systems coupled via the Morse potential. In our proof we switch to the velocity-acceleration coordinates and focus on the study of dissipativity properties for a more general class of Liénard systems $\ddot x_k = -\mathbb F_k(x_k)\cdot \dot x_k -\sum_{m} a_{k,m}x_m$, $k=1,\ldots,n$, $\mathbb F_k(x) = \nabla F_k(x)$ with $F_k: \mathbb R^d\rightarrow \mathbb R^d$ given by $F_k(x) = p_k(|x|)x$.

math.DS↗

arXiv2021

On the stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems

In this paper, we study the dynamics of a system of $n$ coupled, self-propelled particles: $\ddot r_k = (\alpha-\beta |\dot r_k|^2)\dot r_k - \frac{\gamma}{n}\sum_{m=1}^n(r_k-r_m)$, $r_k\in \mathbb R^2.$ Numerical experiments indicate that, for a large set of initial conditions, after an initial drift, the center of mass converges to a stationary point, with each particle eventually rotating around it with constant angular velocity. The distribution of particles on the circle need not be uniform. These limit configurations, where all particles rotate in the same direction, are termed {\it rotating states} . We prove that rotating states are stable and that every solution that starts sufficiently close, asymptotically approaches a rotating state, exponentially fast if $n$ is odd, or at a rate that may be exponential or $\frac{1}{\sqrt t} $ if $n$ is even. The proof uses a new approximation technique for the flow on the center manifold in the presence of non-isolated fixed points.

math.DS↗