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A Proof On Arnold Chord Conjecture

In this article, we first give a proof on the Arnold chord conjecture which states that every Reeb flow has at least as many Reeb chords as a smooth function on the Legendre submanifold has critical points on contact manifold. Second, we prove that every Reeb flow has at least as many close Reeb orbits as a smooth round function on the close contact manifold has critical circles on contact manifold. This also implies a proof on the fact that there exists at least number $n$ close Reeb orbits on close $(2n-1)$-dimensional convex hypersurface in $R^{2n}$ conjectured by Ekeland.