also that the cases $\alpha>0$ and $\alpha<0$ seem to be fundamentally different; we do not have a way of conjugating one with the other.

Nonlinear parabolic maps {#secnonlinexample}
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One can also construct nonlinear parabolic maps that display similar structure in their maximal invariant sets as the piecewise linear case. Consider the two parameter family of the maps on the torus defined by $$\label{nonlin}
\begin{array}{l}
x'=x+ \beta(e^{\gamma (y-x)} -1)~~(\bmod ~1)\\
y'=y+ \beta(e^{\gamma (y-x)} -1)~~(\bmod ~1)\\
\end{array}$$ For any non-zero values of the real parameters $\beta$ and $\gamma$ the Jacobian $\partial(x',y')/\partial(x,y)$ is constant and equal to one. Moreover, for any point $(x,y)$ the trace of the map $t$ is equal to $2$, which makes the above map similar to the linear case (\[eqmap\]). Observe that the diagonal of the unit square $y=x$ is invariant with respect to this map. This map acting on the plane transforms the unit square into a set confined between four exponential functions. After folding it back into the unit square some fragments of it will overlap as (\[nonlin\]) is typically noninvertible.

(6,6)(0,0) (0.5,0.5) (0.2,0.5) (0.45,0.25) (0.25,5.5) (5.5,0.25) (0.5,0.5)[(5.0,5.0)]{} (0.5,5.5)[(0,1)[0.25]{}]{} (5.5,0.5)[(1,0)[0.25]{}]{} (0.45,5.85) (5.8,0.45)

\[fignonlin\]

Figure \[fignonlin\] shows the invariant set of the nonlinear map in the case 