$\beta=\gamma=1$. There is numerical evidence that the maximal invariant set has positive volume in this case even though the Jacobian changes between being a rational and an irrational parabolic map at different points.

Discussion {#secdiscuss}
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In this paper we have considered some basic properties of the maps (\[eqmap\]) in the parabolic area-preserving regime, and show a surprising degree of sensitivity of the dynamical behaviour on rationality of the parameters $\alpha$ and $A$. Our investigations raise some intriguing questions concerning how the Lebesgue measure of the maximal invariant set varies with parameters. For several examples do we have analytical values of this measure. For general rational parabolic maps we do have a result that gives upper and lower bounds for this measure. However, these bounds become weak as one examines higher denominator rational mapsand do not easily give insight into the measure for irrational parabolic maps.

It would be very informative to understand the structure of invariant measures of the map (\[eqmap\]). Note that $\ell(.)$ (Lebesgue measure) restricted to $X^+$ is invariant under this mapping but it is not ergodic. In particular, it may be the case that this restriction is trivial in which case empirical measures can still be 