On approximation by Function having a strong Entropy Point

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: Conn_C (which is a subfamily of the class Conn introduced in [Korczak-Kubiak. E.. Paw- lak. R.J.: Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions. Czechoslovak Math. J. 63 (2013). 679-700]). The main result of the paper Is a theorem saying that for any function ƒ ∈ Conn_C and any point x₀ ∈ Fix(ƒ) there exists a ring R ⊂ Conn_C containing function ƒ and in the intersection of any “graph neighbourhood of ƒ” and “neighbourhood of ƒ in topology of uniform convergence”, one can find functions ξ,Ψ ∈ R having a strong entropy point y₀ located close to the point x₀ and being a discontinuity point of the function ξ and a continuity point of the function Ψ.