The topological entropy versus level sets for interval maps
We answer affirmatively Coven’s question [PC]: Suppose f : I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2? 0. Introduction. The aim of this paper is to estimate the topological entropy of an interval map knowing the cardinalities of its level sets. The topological entropy as a numerical measure for the complexity of a dynamical system is deeply studied in various contexts. One direction of possible research concerns the connection between entropy and level sets for a continuous (differentiable) map. Several interesting results have already been found—see for instance [Bo], [Ly], [MP], [MS]. However, many questions remain open even for interval maps. Our goal is to show one very particular result (Theorem 3.3), the proof of which is neither immediate nor easy. The solution is based on known strong results (Theorems 1.1 and 1.3) concerning the topological entropy and symbolic dynamics. We suppose that our proof can partially explain some difficulties that we meet when similar questions are considered in other topological dynamics. Let I = [a, b] be a closed real interval and let L2(I) be the set of all continuous functions mapping I into itself that satisfy the condition (1) ∀y ∈ I: card f−1(y) ≥ 2. We show the following. Theorem 3.3. The topological entropy of f ∈ L2(I) is greater than or equal to log 2. The problem of entropy of maps from L2(I) has been stated by E. M. Coven in [PC]. The idea why the entropy of any map f from L2(I) should 2000 Mathematics Subject Classification: 37E05, 37B40.