On the entropy of Darboux functions

Abstract

Abstract. We prove some results concerning the entropy of Darboux (and almost continuous) functions. We first generalize some theorems valid for continuous functions, and then we study properties which are specific to Darboux functions. Finally, we give theorems on approximating almost continuous functions by functions with infinite entropy.

On ∞-entropy points in real analysis

“…However, in many situations it is advantageous to consider points "close to fixed points" (in the sense that fixed points of the function lie arbitrarily close to a given point, e.g. almost fixed points and Theorem 3.2 [22]). It should be noted that the consideration of almost fixed points makes sense only for discontinuous functions.…”

“…Using the properties of almost fixed points (see [22]) we get immediately that for any f ∈ DB 1 if x 0 ∈ aFix(f ), then x 0 ∈ pFix(f ). We can prove even more.…”

“…Sharkovskiȋ's theorem [27], or basic theorems regarding entropy [9]. Starting from the paper [22] (see also [21]), there is a new trend of considering results which are not the generalizations of properties known for continuous functions. The results presented in this paper correspond with this field of research.…”

“…Simultaneously, analysis of different examples of functions (not only of real variables -see [23]) has resulted in an interesting observation that entropy of a function may be focused at the point ( [22,23]). This enabled us to define the notion of the entropy point of a function ( [23]) in quite a general spaces.…”

On some subclasses of the family of Darboux Baire 1 functions

“…Many papers which appeared during the last few years contain results concerning Darboux-like functions in relation to dynamical systems (see [3,8,[20][21][22][23]). …”