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We prove that the Benjamin--Ono equation on the torus is globally in timewell-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > -1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent$s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness onthe torus. The obtained solutions are almost periodic in time. Furthermore, weprove that the traveling wave solutions of the Benjamin-Ono equation on thetorus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > -1/2$. Novel conservation laws and a nonlinear Fourier transform on$H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into theproofs of these results.

Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution