On the topological dynamics and phase-locking renormalization of Lorenz-like maps

Abstract

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class if Lorenz-like maps considered as a map of the circle of degree on with one doscontinuity. In a second step we will consider the subclass of the Lorenz-like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the setof renormalizable maps with rotation interval degenerate to a rational number, that is, of phase-locking renormalizable maps. This characterization is given byshowing the equivalence between the geometric renormalization procedure and the combinatorial one (which is expressed in terms of an *- like product defined in the set of kneading invariants). Finally, we will prove the existence, at a combinatorial level, of periodic points of all periods for the renormalization map.