Axiomatic definition of the topological entropy on the interval (Q1396533)

Let \(C(I)\) be the space of all \(I \to I\) continuous functions, \(I\) real compact interval . Two groups of axioms on a mapping \(A:C(I) \to [0,\infty]\) characterizing topological entropy are given. A map \(A:C(I) \to [0,\infty]\) is said to characterize chaos if for any \(f \in C(I)\) \(A(f)>0\) is equivalent to positive topological entropy of \(f\). A third group of axioms given in the paper is sufficient for a map \(A:C(I) \to [0,\infty]\) to characterize chaos. Neccessity of individual axioms is discussed.