Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family

Abstract: We introduce a renormalization model which explains how the behavior of a discrete-time continuous dynamical system changes as the dimension of the system varies. The model applies to some two-dimensional systems, including H\'enon and Lozi maps. Here, we focus on the orientation preserving Lozi family, a two-parameter family of continuous piecewise affine maps, and treat the family as a perturbation of the tent family from one to two dimensions. First, we give a new prove that all periodic orbits can be classified by using symbolic dynamics. For each coding, the associated periodic orbit depends on the parameters analytically on the domain of existence. The creation or annihilation of periodic orbits happens when there is a border collision bifurcation. Next, we prove that the bifurcation parameters of some types of periodic orbits form analytic curves in the parameter space. This improves a theorem of Ishii (1997). Finally, we use the model and the analytic curves to prove that, when the Lozi family is arbitrary close to the tent family, the order of periodic orbit creation reverses. This shows that a forcing relation (Guckenheimer 1979 and Collett and Eckmann 1980) on orbit creations breaks down in two dimensions. In fact, the forcing relation does not have a continuation to two dimensions even when the family is arbitrary close to one dimension.