We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved in [KMS16]. This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the $b$-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on $b$-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.
Analytic Liouville Integrable Geodesic Flow Has Zero Entropy
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems
Centre for Automation and Robotics, Spanish National Research Council (CSIC)（西班牙国家研究理事会自动化与机器人中心）；Institute of Mathematical Sciences, Spanish National Research Council (CSIC)（西班牙国家研究理事会数学科学研究所）；Royal Academy of Exact, Physical and Natural Sciences（皇家精确、物理与自然科学学院）；Department of Quantitative Methods, CUNEF University（CUNEF大学定量方法系）；Institute of Mathematics of the Polish Academy of Sciences (IM PAN)（波兰科学院数学研究所）
