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A note on the Definition of a僕imit Set |
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PP: 1929-1932 |
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Author(s) |
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Francisco Balibrea,
Juan L.G. Guirao,
Marek Lampart,
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Abstract |
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| 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僕imit 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僕imit set is equal to the whole space X; there is a complete negative
trajectory of x whose alpha僕imit set is equal to a fixed point; there is a complete negative trajectory of x whose alpha僕imit set is equal
to X. |
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