Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem

Abstract

We construct a kneading theory à la Milnor - Thurston for Lozi mappings (piecewise affine homeomorphisms of the plane). As a two-dimensional analogue of the kneading sequence, the pruning front and the primary pruned region are introduced, and the admissibility criterion for symbol sequences known as the pruning front conjecture is proven under a mild condition on the parameters. Using this result, we show that topological properties of the dynamics of the Lozi mapping are determined by its pruning front and primary pruned region only. This gives us a solution to the first tangency problem for the Lozi family, moreover the boundary of the set of all horseshoes in the parameter space is shown to be algebraic.