Topological entropy of compact subsystems of transitive real line maps

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 ([0,1), respectively). It is proved that if f is topologically transitive, then the best lower bound of ent(f) is (log3, respectively) and it is not attained. This solves a problem posed by Canovas [Topological entropy of continuous transitive maps on the real line, Dyn. Syst. 24(4) (2009), pp. 473-483]. It is also shown that for non-compact spaces the specification property is no longer a conjugacy invariant and there are mixing maps of the open and half-open interval without the specification property.