Maslov-type (L,P)-index and subharmonic P-symmetric brake solutions for Hamiltonian systems

This paper introduces a novel iteration inequality for the Maslov-type $(L,P)$-index of iterated symplectic paths. Here, P is a fixed 2n-dimensional symplectic and orthogonal matrix satisfying $P^m=I$. These advancements in index theory are then applied to investigate the multiplicity of subharmonic solutions in Hamiltonian systems exhibiting dihedral equivariance with period mτ. Notably, a criterion of geometric distinction is established for two subharmonic P-symmetric brake orbits with periods kmτ and lmτ within the set $\left\{km\tau \right|k\equiv 1\text{ (mod }m\left)\right\}$. This criterion is based on a lower bound estimate for the ratio $l/k$. Specifically, for odd k, the lower bound must be not less than $(\frac{1}{2}\dim \ker (P-I)+2)m+1$, while for even k, it must be not less than $(\frac{1}{2}\dim \ker (P-I)+n+2)m+1$.