Invariance of Gibbs Measures on the Phase Space of the Periodic Ablowitz-Ladik System

In this paper, we analyze the dynamics of the periodic defocusing Ablowitz-Ladik system, a discretization of the nonlinear Schrödinger equation. In particular, we investigate the conservation laws and invariance properties of Gibbs measures for the AL system by leveraging its Hamiltonian structure and integrability. We first establish the existence of a maximal family of conservation laws which underpin its dynamics. Building on these conservation laws, we construct Gibbs measures associated with the AL system through statistical ensembles weighted by its conserved energies. Utilizing techniques from integrable systems theory and symplectic geometry, we prove that these measures remain invariant under the AL flow, thereby bridging the dynamics of the discrete system with its statistical equilibrium properties.