Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $

In this paper, we provide new index estimations and prove that for any $P$-symmetric compact convex hypersurface $\Sigma$ in $\mathbb{R}^{2n}$, i.e. $x\in\Sigma$ implies $Px\in\Sigma$ with a certain orthogonal symplectic matrix $P$, there are at least $[\frac{3n}{4}]$ closed characteristics on $\Sigma$. Provided there exist an integer $m>2$ such that $$P^m=I_{2n},$$ and there exist only one $\theta\in(0,\pi]$ s.t. $e^{\sqrt{-1}\theta}\in\sigma(P)$ which satisfies $$S^+_P(e^{\sqrt{-1}\theta})=S^-_P(e^{\sqrt{-1}\theta}),$$ where $S^{\pm}_P(\omega)$ are the splitting numbers of $P$ at $\omega\in \mathbf{U}:=\{z\in\mathbb{C},|z|=1\}$.

[1]  Duanzhi Zhang.  $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R 2n , 2012 .

[2]  Wei Wang,et al.  Closed trajectories on symmetric convex Hamiltonian energy surfaces , 2009, 0909.3564.

[3]  Shanshan Tang,et al.  Maslov (P, ω)-Index Theory for Symplectic Paths , 2015 .

[4]  Yiming Long,et al.  Closed characteristics on compact convex hypersurfaces in $\R^{2n}$ , 2001 .

[5]  Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces , 2018 .

[6]  Y. Long,et al.  Multiple brake orbits in bounded convex symmetric domains , 2006 .

[7]  A. Szulkin.  Morse theory and existence of periodic solutions of convex hamiltonian systems , 1988 .

[8]  I. Ekeland.  Convexity Methods In Hamiltonian Mechanics , 1990 .

[9]  Alan Weinstein,et al.  Periodic Orbits for Convex Hamiltonian Systems , 1978 .

[10]  Paul H. Rabinowitz,et al.  On the existence of periodic solutions for a class of symetric Hamiltonian systems , 1987 .

[11]  A. Szulkin.  An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems , 1989 .

[12]  Chun-gen Liu,et al.  Multiplicity of closed characteristics on symmetric convex hypersurfaces in R , 2022 .

[13]  Wei Wang.  Closed characteristics on compact convex hypersurfaces in $\R^8$ , 2013, 1305.4680.

[14]  Yiming Long,et al.  Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$ , 2001 .

[15]  Shanzhong Sun,et al.  Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit , 2009 .

[16]  Chun-gen Liu,et al.  Iteration theory of Maslov-type index associated with a Lagrangian subspace for symplectic paths and Multiplicity of brake orbits in bounded convex symmetric domains , 2009 .

[17]  Joel W. Robbin,et al.  The Maslov index for paths , 1993 .

[18]  Y. Long,et al.  Closed characteristics on partially symmetric compact convex hypersurfaces in R2n , 2004 .

[19]  I. Ekeland,et al.  Convex Hamiltonian energy surfaces and their periodic trajectories , 1987 .

[20]  P. Rabinowitz,et al.  Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems , 1977 .

[21]  Y. Long.  Index Theory for Symplectic Paths with Applications , 2002 .

[22]  Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$ , 2016 .