Monotonicity of the Lozi Family and the Zero Entropy Locus

Ishii and Sands showed the monotonicity of the Lozi family ${\mathcal{L}}_{a,b}$ in a ${\mathcal{C}}^1$ neighborhood of a-axis in the $a$-$b$ parameter space. We show the monotonicity of the entropy in the vertical direction around $a=2$ and in some other directions for $1<a\le 2$. Also we give some rigorous and numerical results for the parameters at which the Lozi family has zero entropy.