This survey is devoted to investigations concerning topological, algebraic, and combinatorial characteristics as well as metric invariants for arbitrary groups of homeomorphisms of the line and the circle. Relationships between these characteristics are established, the most important metric invariants are studied (in the form of invariant, projectively invariant, and ω-projectively invariant measures), and the main 'obstructions' to the existence of metric invariants of this kind are described.

https://iopscience.iop.org/article/10.1070/RM2004v059n04ABEH000758/pdf

Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants

К вопросу о классификации групп гомеоморфизмов $\mathbb R$, сохраняющих ориентацию. III. $\omega$-проективно-инвариантные меры@@@On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. III. $\omega$-projectively invariant measures

The presence of a trapped mode in an asymmetric waveguide with slightly deformed hard walls—in particular, in a cranked waveguide—is revealed. The perturbation profile is subject to certain integral conditions. The result is obtained by analyzing the augmented scattering matrix.

Trapped waves in a cranked waveguide with hard walls

We study the discrete spectrum of the Laplace operator with the Dirichlet boundary conditions in a plane waveguide made of a band of unit width and half-band of width h perpendicular to it. The existence of critical width h
 * ∈ (1,2] is established: in the case of h ∈ (0, h*), the discrete spectrum σ
 of the problem contains only one eigenvalur λ
 1
 , but σ
 = ⊘ at h ≥ h
 *. The asymptotics of the eigenvalue λ
 1
 at h → + 0 is found.

Trapped modes in a T-shaped waveguide

/pdf/k-voprosu-o-klassifikatsii-grupp-gomeomorfizmov-mathbb-r-3dktvxrmzi.pdf

К вопросу о классификации групп гомеоморфизмов $\mathbb R$, сохраняющих ориентацию. II. Проективно-инвариантные меры@@@On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. II. Projectively-invariant measures

/pdf/k-voprosu-o-klassifikatsii-grupp-gomeomorfizmov-mathbb-r-37805js2jx.pdf

К вопросу о классификации групп гомеоморфизмов $\mathbb R$, сохраняющих ориентацию. I. Инвариантные меры@@@On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. I. Invariant measures

/pdf/ob-analogakh-alternativy-titsa-dlia-grupp-gomeomorfizmov-nhqnd7uv71.pdf

Об аналогах альтернативы Титса для групп гомеоморфизмов окружности и прямой@@@On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

/pdf/gruppy-gomeomorfizmov-priamoi-i-okruzhnosti-topologicheskie-2y81yzwxqf.pdf

Группы гомеоморфизмов прямой и окружности. Топологические характеристики и метрические инварианты@@@Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants

Лева Андреевич Бекларян

Papers

К вопросу о классификации групп гомеоморфизмов $\mathbb R$, сохраняющих ориентацию. I. Инвариантные меры@@@On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. I. Invariant measures

Об аналогах альтернативы Титса для групп гомеоморфизмов окружности и прямой@@@On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

Группы гомеоморфизмов прямой и окружности. Топологические характеристики и метрические инварианты@@@Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants