## 2 The Lozi Chaotic Map
### Advanced Dynamical Properties of the Dissipative Map (\(\,\mid b\mid\,<1\))
In addition bifurcation studies, Collet and Levy [20] considered ergodic properties of the Lozi map, which they considered as an intermediate stage between the Axiom \(A\) dynamical systems and more complicated systems such as the Henon map. They constructed its Bowen-Ruelle measure and derived several of its properties, which are similar to those of an axiom \(A\) system. A proof of the existence of Sinai-Bowen-Ruelle measures (SBR measures) for this map was provided by Rychlik [21], who additionally proceed that the number of SBR measures is finite. The geometric structure of the chaotic attractor was explored by Cao and Liu [22], who proved the following:  
**Proposition 1**. : _If the parameters \((a,b)\) satisfy the Misiurewicz conditions (4), then the strange attractor \(\Lambda_{a,b}\) possesses the following properties:_  
* _The union of the transversal homoclinic points and weak transversal homoclinic points is dense in_ \(\Lambda_{a,b}\)__
* _All periodic points are hyperbolic_
* _The set of periodic points forms a dense set in_ \(\Lambda_{a,b}\)__
* _Any two hyperbolic points forms a transversal heteroclinic cycle or a weak transversal heteroclinic cycle_  
Other statistical, hyperbolic, ergodic, and topological properties of the Lozi attractor can be found in Afraimovich et al. [23]. The symbolic dynamics of this map have been extensively studied. In 1991, Zheng [24] described details of this dynamics. To this end, they assigned two families of symbolic sequences for two groups of lines in the phase plane, allowing the order of symbolic sequences to be defined, and derived the ordering rules. Misiurewicz and Stimac [11], in a more detailed study, proved that the set of kneading sequences they introduced determines its symbolic dynamics. They defined two other equivalent approaches as well. The important works of Ishii [15; 25], Sand [26], and de Carvalho and Hall [27] in this field of research can be mentioned here as well. In [15], Ishii constructed a kneading theory a la Milnor-Thurston and showed that topological properties of the dynamics of the Lozi map are determined by its pruning front and primary pruned region only. This allowed a solution to the first tangency problem for the Lozi family to be found. In addition, the boundary of the set of all horseshoes in the parameter space was shown to be algebraic. In [25], the partial monotonicity of the topological entropy and of bifurcations near horseshoes was proved as an application of this result. Upper and lower bounds for the Hausdorff dimension of the Lozi attractor were provided in terms of parameters. In [27], recent results on pruning theory were provided, concentrated on prunings of the horseshoe. The monotonicity of the Lozi family when the Jacobian determinant is close to zero was proved in [26]. The main ingredients of the proof therein are the \"pruning pair method\" and a detailed analysis of the parameter dependence of the kneading invariant of the tent-map family. Wormell [28] defined a new property of chaotic attractors, namely, conditional mixing, and conjectured:  
**Conjecture 1**. : _For generic Lozi parameters \((a,b)\) and almost all Lebesgue \(x_{0}\in R\), the Lozi map has conditional mixing with respect to level curves \(x=x_{0}\), that is, the measures \(\rho(\cdot|x_{0})\)._  
Strong and direct numerical evidence has been presented in favor of this conjecture. To obtain such a result rigorously, novel algorithms were developed in [28] that quantify the associated numerical error. Borinski and Stimac [29] showed that Henon-like and Lozi-like maps (see [2] for a definition of such maps) on their strange attractors are conjugate to natural extensions of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To conclude this section, I refer to the general survey of Anischenko et al.