Lozi maps

The Lozi family consists of the maps

The attractor for Lozi's original parameters: $a = 1.7, b = 0.5$.

(1)

\begin{align} \mathcal{L}_{a, b} : \binom{x}{y} \longmapsto \binom{1-a|x| + by}{x}, \end{align}

where $a, b\in {\mathbb R}$ are parameters. Some authors prefer the definition

(2)

\begin{align} \mathcal{\tilde{L}}_{a, b} : \binom{x}{y} \longmapsto \binom{1-a|x|+y}{b x} \end{align}

which has the advantage of clearly degenerating to a one-dimensional tent-map when $b = 0$. If $b\neq 0$ then there is no real difference between the two definitions since one can be transformed into the other by a linear coordinate change. We will use the first definition.

As long as $b\neq 0$, which is the interesting case, $\mathcal{L}_{a, b}$ is a piecewise affine homeomorphism of ${\mathbb R}^2$. If $b=0$ ("the one-dimensional case") then $\mathcal{L}_{a, 0}$ is equivalent in a slightly twisted way to the tent-map $x \mapsto 1-a|x|$ and has topological entropy $\log(a)$ if $1 \leq a \leq 2$.

At the opposite extreme, if $b = \pm 1$ then $\mathcal{L}_{a, b}$ is area preserving. To see why this is the "most two-dimensional case", consider the Jacobian determinant of $\mathcal{L}_{a, b}$, which is equal to $-b$ everywhere. Since the Jacobian is zero in the one-dimensional case, the size of $b$ can be thought of as measuring how two-dimensional $\mathcal{L}_{a, b}$ is. The bigger the better you might think, but there is a twist: the inverse of $\mathcal{L}_{a, b}$ is conjugate to $\mathcal{L}_{a/|b|, 1/b}$ by an affine coordinate change, so maps with $|b| > 1$ can be considered to be maps with $|b| < 1$ in disguise. A map with $b = \pm 1$ is affinely conjugate to its own inverse.

The Lozi family was introduced as a simpler version of the Hénon family

The attractor for Hénon's original parameters: $a = 1.4, b = 0.3$.

(3)

\begin{align} \mathcal{H}_{a, b} : \binom{x}{y} \longmapsto \binom{1-a x^2+y}{b x}. \end{align}

In the one-dimensional setting, quadratic maps with positive entropy are always semi-conjugate to the tent-map with the same topological entropy (if the critical point of the tent-map is not periodic then the semi-conjugacy is a conjugacy). Unfortunately the analogous statements relating Hénon and Lozi maps are far from being true (need a reference and discussion!). The similarities between Hénon and Lozi maps are nonetheless striking, so perhaps something can still be said.

Area preserving orientation reversing Lozi map $L_{1.5,1}$. Each colour is the orbit of a single point.

Devaney's cookie cutter: $L_{-1,-1}$. Each connected block of colour is the orbit of a single point.

Area preserving Lozi map $L_{1,-1}$. Each connected block of colour is the orbit of a single point.

Area preserving Lozi map $L_{1.01,-1}$. Each connected block of colour is the orbit of a single point.

Area preserving Lozi map $L_{1.05,-1}$. Each connected block of colour is the orbit of a single point.