Dominant Axis Theorem and the Area Preserving Lozi Map

In the family of the area preserving Hénon maps (the Hénon maps), the mapping function is quadratic. Replacing the quadratic function with a piecewise linear function, we obtain the area preserving Lozi map (the Lozi map). For the Hénon map, the elliptic periodic orbits appearing through rotation bifurcation of the elliptic fixed point have one orbital point on the particular axis, i.e., the dominant axis. Thus, the dominant axis theorem holds for the Hénon map. For the Lozi map, the dominant axis theorem does not hold. We make clear the reasons from the study of bifurcations. For the Lozi map, a new theorem instead of the dominant axis theorem is obtained.