COLLOQUIUM MATHEMATICUM VOL. 116 2009 NO. 2 ON THE ENTROPY OF DARBOUX FUNCTIONS BY RYSZARD J. PAWLAK (Łódź) 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 en- tropy. Introduction. In the classical theory of dynamical systems one usually assumes that all the functions under consideration are continuous. However, some investigations lead to considering Darboux functions or almost contin- uous functions (cf. e.g. a question raised by W. Transue, cited in [10]); a discussion of this topic can be found in [20]. There are a lot of recent papers dealing with dynamical systems generated by Darboux-like functions (e.g. [7], [10], [16], [18], [21]). The main aim of the current one is to give some results on the entropy of Darboux or almost continuous functions. Our direct inspiration was M. Čiklová’s paper [7]. She generalizes a cer- tain theorem valid for continuous functions to the case of functions whose graph is a connected G δ set (in particular, such functions have the Darboux property). We mainly aim at the properties specific to dynamical systems generated by Darboux functions. But we also extend several classical results, regarding them as important tools (see Section 2). In Section 3 we define almost fixed points of a Darboux function. This notion is characteristic for discontinuous Darboux functions. We investigate the fundamental properties of Darboux (or Darboux-like) functions having at least one almost fixed point. In Section 4 we consider approximation of an arbitrary almost continuous function by almost continuous functions having almost fixed points (or having infinite entropy). The paper is completed by an open problem concerning the relationship between the entropy of a function and the set of periodic points of this function. 2000 Mathematics Subject Classification : 37B40, 26A18, 54H20, 26A15, 37E15. Key words and phrases : topological entropy, Darboux function, almost fixed point, almost continuity, periodic point, m-horseshoe, turbulent function, T -approximation. DOI: 10.4064/cm116-2-7 [227] c  Instytut Matematyczny PAN, 2009