The topological entropy of the transformationx ↦ax (1−x)

A well-known conjecture about the transformationTa:x ↦ax (1−x) on [0, 1], where 2≤a≤4, says that the mapa ↦htop (Ta) is monotone. In this paper we show that this is connected with a property of the polynomialsPk (t) (4≤t≤8) given byP0 (t)=0 andPk+1(t)=(t−Pkt)2)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a∈[2,4]:htop (Tc), ifca}.