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In this paper, we consider the cubic fourth-order nonlinear Schr\"odingerequation (4NLS) under the periodic boundary condition. We prove two results.One is the local well-posedness in $H^s$ with $-1/3 \le s < 0$ for the Cauchyproblem of the Wick ordered 4NLS. The other one is the non-squeezing propertyfor the flow map of 4NLS in the symplectic phase space $L^2(\mathbb{T})$. Toprove the former we used the ideas introduced in [Takaoka and Tsutsumi 2004]and [Nakanish et al 2010], and to prove the latter we used the ideas in[Colliander et al 2005].

Periodic fourth-order cubic NLS: Local well-posedness and Non-squeezing property