Homoclinic Tangencies in Unimodal Families with Non-constant Topological Entropy

Let C r ((0; 1]) denote the metric space of C r self-maps of 0; 1] and C s;r ((0; 1]) denote the metric space of C s families of maps in C r ((0; 1]) with the parameter space 0; 1]. Let Hs;r be the unimodal families with non-constant topological entropy in C s;r ((0; 1]). We show that for s 0, r 2, there is an open and dense subset Gs;r of Hs;r such that each family in Gs;r has a map with a homoclinic tangency. If s 0, r = 1, only density must hold. As the key step, with the Decomposition Theorems of Jonker & Rand and Blokh, and the Semiconjugacy Theorem of Milnor & Thurston, for s 0, r 2, we show that a generic family ff g 20;1] 2 Hs;r has a map which is arbitrarily C r-close to a map with a homoclinic tangency. This lemma could be restated as follows: for a unimodal map f 2 C r ((0; 1]), r 2, with only one critical point, which is the nondegenerate turning point, either some neighborhood of f in C r ((0; 1]) has constant topological entropy, or every neighborhood of f in C r ((0; 1]) has some map with a homoclinic tangency.