Multiplicity and stability of closed characteristics on compact convex P-cyclic symmetric hypersurfaces in \bf R²ⁿ

2021/02/13 by Liu, Hui
#34C25 #37J45 #58E05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.06832

Abstract

Let Σ be a compact convex hypersurface in \bf R²ⁿ which is P-cyclic symmetric, i.e., x∈ Σ implies Px∈Σ with P being a 2n×2n symplectic orthogonal matrix and satisfying P^k=I_2n, ker(P^l-I_2n)=0 for 1≤ l< k, where n, k≥2. In this paper, we prove that there exist at least n geometrically distinct closed characteristics on Σ, which solves a longstanding conjecture about the multiplicity of closed characteristics for a broad class of compact convex hypersurfaces with symmetries(cf.,Page 235 of \citeEke1). Based on the proof, we further prove that if the number of geometrically distinct closed characteristics on Σ is finite, then at least 2[(n)/(2)] of them are non-hyperbolic; and if the number of geometrically distinct closed characteristics on Σ is exactly n and k≥3, then all of them are P-cyclic symmetric, where a closed characteristic (τ, y) on Σ is called P-cyclic symmetric if y(\bf R)=Py(\bf R).

Related

• Multiplicity of closed characteristics on P-symmetric compact convex hypersurfaces in ℝ²ⁿ
• Stability of closed characteristics on symmetric compact convex hypersurfaces in \R²ⁿ
• Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in \bf R²ⁿ
• Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces
• Existence of closed characteristics on compact convex hypersurfaces in \R²ⁿ