A note on the Definition of α–limit Set

Many phenomena coming from the biology, economy, engineering are modeled using discrete dynamical systems. The concept of backward orbit is an essential concept for understanding the dynamics of the system. In the literature various definitions of the concept of the alpha–limit point (respectively set) have been historically used. The aim of this paper is to analyze the forcing relationships between them via the proof of the valid relationships and the construction of counterexamples for the converse situation in order to clarify the scenario for the computation of these objects. Moreover, we present a discrete dynamical system (X , f ) with the following paradoxical behavior: for every point x ∈ X , its alpha–limit set is equal to the whole space X ; there is a complete negative trajectory of x whose alpha–limit set is equal to a fixed point; there is a complete negative trajectory of x whose alpha–limit set is equal to X .