I will briefly discuss three important central concepts in the theory of dynamical systems. The first part of the talk is about introducing some basic notions in dynamics such as the omega-limit set, map-connectedness, fixed point property, etc. Then, I focus on a particular class of two-dimensional dynamical systems. Omega-limit sets are considered to be a powerful tool to understand a system. Dynamical systems of higher dimensions has not been completely studied so far. I introduce a particular two-dimensional class of dynamics, called triangular maps of the unit square, and describe their omega-limit sets in terms of the map-connectedness. The second part of the discussion is about the property of chain transitivity. The last part is dedicated to the fixed-point property. In particular, I demonstrate that a particular set of some compact sets has the fixed-point property.
Bienvenue à tous.