Topological entropy of compact subsystems of transitive real line maps

Abstract. For a continuous map f from the real line (half-open interval [0, 1)) into itself let ent(f) denote the supremum of topological entropies of f |K, where K runs over all compact f-invariant subsets of R ([0, 1), respectively). It is proved that if f is topologically transitive, then the best lower bound of ent(f) is log 3 (log 3, respectively) and it is not attained. This solves