Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([-1, 1], f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([-1, 0], (f) over tilde) be the quotient map, given by the equivalence relation x similar to -x, the dynamic of (f) over tilde is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f. Moreover we show that in general ([-1, 0], (f) over tilde) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).