Zeros of the kneading invariant and topological entropy for Lorenz maps

Abstract

If is a unimodal map, then its topological entropy is related to the smallest positive zero s of a certain power series (the kneading invariant of f) by . Moreover, it is implicit in the results of Jonker and Rand that for each positive entropy basic set in the renormalization decomposition of the non-wandering set of f, there is a real zero of the kneading invariant such that . Here we prove a similar result for Lorenz maps. In contrast to the unimodal case, it is possible for two basic sets in the renormalization decomposition of the non-wandering set of a Lorenz map to have the same entropy, and we show that in this case there is a corresponding double zero of the kneading invariant.