On two notions of fuzzy topological entropy

We explore the notion of fuzzy topological entropy when different definitions of fuzzy compactness are considered. We prove that the recent definitions by Tok (2005) [23] and by Uzzal Afsan and Basu (2011) [24] are not the most appropriate since they always display zero entropy. Furthermore, we give a simple proof of the bridge result, which states that topological entropy agrees with fuzzy topological entropy when it is defined by using Lowen's definition of compactness. Consequently, many natural properties of the fuzzy topological entropy (such as monotonicity) are obtained as direct corollaries. The particular case of interval maps is also briefly discussed.