Level Sets Lemmas and Unicity of Critical Point of Invariants, Tools for Local Stability and Topological Properties of Dynamical Systems - Bastien, Guy; Rogalski, Marc (Sarajevo Journal of Mathematics, 2024)
We prove that the level curves of some differentiable functions of two variables with unique critical point are diffeomorphic to the circle T, and show how this result can be used in the study of local stability of dynamical systems in dimension 2 with invariant function, without using the Hessian. We extend the results to the level sets of an invariant function of dynamical systems, with a synthesis exposition of examples of improvements of previously studied order q difference equations with invariant. In fine we present some differential tools for the study of the topological nature of invariant level sets in dimension at least three. 2000 Mathematics Subject Classification. 37E75, 39A20, 39A30