Abstract.
For any Hénon map $H$, you can add a divisor $X$ at infinity to $C^2$, together with two non-analytic points, to make a compact dynamical system. If you quotient the set of the points that escape to infinity under forward iteration of $H$, together with $X$, by the action of $H$, you obtain a compact analytic surface $S^+$ (belonging to Kodaira’s class VII). You can do the same for $H^{-1}$, obtaining a surface $S^-$. These two surfaces have a complicated fractal subset, and you can cut out one and glue in the other, to turn one surface into the other.