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 of Lorenz--like maps considered as a map of the circle of degree one with one discontinuity. 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 set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase--locking renormalizable maps. This characterization is given by showing 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.