insight - Dynamical Systems

### アメナブル群の自由作用における URP、比較、平均次元、および鋭いシフト埋め込み可能性

アメナブル群の自由作用において、URP と比較の組み合わせ (URPC) は、平均次元と鋭いシフト埋め込み可能性の間に重要な関係があることを示す。さらに、URPC を特徴づける新しい条件 (FCSB) を導入し、URP と URPC の間の等価性を明らかにする。

### Free Action of Amenable Groups: Uniform Rokhlin Property, Comparison, Mean Dimension, and Sharp Shift Embeddability

For free actions of amenable groups on compact metrizable spaces, the Uniform Rokhlin Property (URP) and the conjunction of URP and comparison (URPC) are shown to be equivalent to technical conditions called property FCSB and property FCSB in measure. These properties are then used to establish a sharp shift embeddability theorem and characterize when the associated crossed product C*-algebras have desirable properties.

### Veech's Theorem on Higher Order Regionally Proximal Relations for Minimal Abelian Dynamical Systems

For a minimal abelian dynamical system (X, G), a point (x, y) is in the regionally proximal relation of order d, RP[d], if and only if there exists a sequence {gn} in Gd and points zε in X for each ε in {0, 1}d{0} such that the limits limn→∞(gn·ε)x = zε and limn→∞(gn·ε)−1z1 = z1-ε hold.

### Finite and Infinite Degree Thurston Maps with a Small Postsingular Set

Thurston maps with at most three singular values and four postsingular values can be characterized by the existence of weakly degenerate Levy fixed curves. The Hurwitz classes of such Thurston maps exhibit interesting properties, including the existence of infinitely many realized and obstructed maps.

### 연속 분절 선형 평면 사상 가족의 불변 그래프와 동역학

a ≥ 0인 경우 모든 궤도는 결국 주기적이며, a < 0인 경우 궤도는 1차원 불변 그래프에 수렴한다. 이 그래프 상에서의 동역학은 매우 다양하며, 양의 엔트로피를 가지는 경우도 있다.

### Dynamics of a Family of Continuous Piecewise Linear Planar Maps

For the family of piecewise linear maps Fa,b(x, y) = (|x| - y + a, x - |y| + b), the dynamics strongly depend on the parameters a and b. When a ≥ 0, all orbits are eventually periodic, while for a < 0, the dynamics is concentrated on one-dimensional invariant graphs that capture the final dynamics of the map.

### Limit Cycles and Chaos in Planar Hybrid Dynamical Systems

This paper studies the family of planar hybrid dynamical systems formed by two linear centers and a polynomial reset map of any degree. It investigates the existence of limit cycles and provides examples of these hybrid systems exhibiting chaotic dynamics.

### 비판적 준원형 사상의 정규화에 대한 쌍곡성

비판적 준원형 사상의 정규화에 대한 쌍곡성을 증명하였다.

### Hyperbolicity of Renormalization of Critical Quasicircle Maps

The paper establishes the hyperbolicity of the renormalization operator acting on the space of critical quasicircle maps with periodic rotation numbers, by constructing a compact analytic corona renormalization operator with a hyperbolic fixed point.

### Aperiodic Points Exist for Outer Billiards on Regular Polygons Except Triangles, Squares, and Hexagons

Outer billiard on any regular N-gon with N > 4 and N ≠ 6 has aperiodic points.