Global behavior of continuous and discrete dynamical systems is an interesting question. For local studies, Poincaré-Bendixon Theorem is a very powerful tool for continuous dynamical systems defined in R2, but in the case of discrete dynamical system is not so. It works if, for instance, we consider the time T-map of a flow given by a vector field. But what happens if the map is not even differentiable in R2? In this case chaotic behavior can appear. First, we will see the relevance of Discrete Markus-Yamabe Conjecture in trying to describe the global dynamics of a fixed point and we will explain when the conjecture is true and when not. We will also stress the cases in R2. Secondly, we will prove the existence of a global asymptotically attacting fixed point for continuous and injective maps of the plane. Finally we will see the relationship between these results and Equivariant Theory in dimension two. Moreover, we will give some applications to Theory of Oscillations and Mathematical Biology via Poincaré map.