Let $C$ be a curve defined by $y = f(x)$, and define the vertical reflection over $C$ to be the map $(x,y) \mapsto (x,y')$, where $y' = 2 f(x) - y$. In other words, the vertical distance from $(x,y)$ to $C$ is the same as the vertical distance from $(x,y')$ to $C$.

Also, you can similarly define the horizontal reflection over a curve of the form $x = g(y)$.

Now, say you have two curves $C_1$ and $C_2$, and let $R_1$ be the horizontal reflection over $C_1$ and $R_2$ the vertical reflection over $C_2$. I am interested in dynamical systems generated by maps of the form $F = R_2 \circ R_1$. 

I found the following PhD thesis involving a specific class of maps of this form: [HERE][1]. This work talks about how if the curves $C_1$ and $C_2$ intersect more than once, then the dynamics has chaotic behavior.

I am however, interested in the converse. If the curves intersect only once, then what extra conditions are needed to ensure the system is *NOT* chaotic? What conditions on $C_1$ and $C_2$ ensure that the dynamics is integrable? In particular I want to know when the orbits are bounded, and lie on simple closed curves.


  [1]: https://link.springer.com/article/10.1007/s12591-016-0304-z