Towards a kneading theory for Lozi mappings. II: Monotonicity of the topological entropy and Hausdorff dimension of attractors
The article is a sequel to \textit{Y. Ishii} [Nonlinearity 10, No.3, 731-747 (1997; Zbl 0908.58015)]. The articles develop symbolic dynamics for Lozi mappings \((x,y) \mapsto (1 - a |x|+ by , x)\), a piecewise linear caricature of the Hénon map. The first article introduces symbolic dynamics and the notions of pruning front and pruning pair. This article discusses some applications of these notions. In particular, it is shown that for each \(b \neq 0\), the topological entropy is a nondecreasing function of \(a\) once \(a\) is larger then some \(a_\ast(b)\) (which is smaller then the infimum of the values of \(a\) for which a full horseshoe exists). Monotonicity properties of bifurcations as \(a\) varies are also discussed. These results have been extended, for small values of \(b\), in \textit{Y. Ishii} and \textit{D. Sands} [Commun. Math. Phys. 198, 397-406 (1998; Zbl 0923.58019)]. The article under review also contains estimates for the Hausdorff dimension of the Lozi attractor in terms of the parameters.
scientific article; zbMATH DE number 1213908