An invariant measure for the equation u_tt-u_xx+u³=0

Numerical studies of the initial boundary-value problem of the semilinear wave equation u_tt-u_xx+u³=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), u_t(t, 0)=u_t(t, 2 π) and initial conditions u(0, x)=u₀(x), u_t(0, x)=v₀(x), where u₀(x) and v₀(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u₀(x), v₀(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above "returning" property.