Simple choreographies of the planar Newtonian $N$-body Problem

In the $N$-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian $N$-body problem with equal masses, $N\ge 3$, there are at least $2^{N-3}+2^{[(N-3)\mathrm{/}2]}$ different main simple choreographies. This confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.