SYMBOLIC DYNAMICS OF BIMODAL MAPS

Our goal in this paper is to introduce an alternative technique to the ones utilized in symbolic dynamics which allows for the simple deduction of formulae for the calculation of the topological entropy. This technique is based on a commutative diagram derived from the study of the homological configurations of graphs associated to bimodal maps of the interval. From the partition of the interval that corresponds to the itinerary of the critical points we introduce 0-1-chains of complexes that translate the dynamical properties in relations of homological type. In [11] it was introduced the kneading-determinant D(t) as a formal power series. On the other hand, in [5] using homological properties we proved a precise relation between the kneading-determinant and the characteristic polynomial of a matrix A associated to the action on 1-chains of the interval with one critical point. In this paper we extend this result to the maps of the interval with two critical points. The advantage of this method is that the formula introduced in proposition 2 allows to write explicitly the characteristic polynomial which is valid for all pairs of sequences of symbols. The study of bimodal maps has found interesting applications in physics and biology [1, 3, 10, 12, 14]. Their mathematical study has drawn the attention of [4, 6, 8, 9, 11, 13]. The introduction of a homological language into these studies was used before by [2].