Abstract

In 1814, while he was a war prisoner in Russia, Poncelet found the following result: known in projective geometry as the Poncelet′s Porism:

"Given one ellipse inside another, if there exists one circuminscribed n-gon (simultaneously inscribed in the outer and circumscribed on the inner), then any point on the boundary of the outer ellipse is the vertex of some circuminscribed n-gon."
More information and a beautifull animation can be viewed in
http://mathworld.wolfram.com/PonceletsPorism.html
The n-gons in the statement of the above result can be seen as periodic orbits of a map defined from the outer ellipse to itself. Indeed, given one ellipse inside another, we can introduce a Map P from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call this type of maps Poncelet′s maps.
Now we can state Poncelet′s Porism in the dynamical systems language: In the two ellipses case and when the rotation number of P is rational there exists a Natural n such that P^n is the identity. In other words, the Poncelet′s map is conjugated to a rational rotation.

In this talk we study general Poncelet′s maps and give several examples of algebraic ovals where the corresponding Poncelet′s map has a rational rotation number and it is NOT conjugated to a rotation. All these cases correspond to integrable maps and we can see that the behaviour of the rotation number function depending on the energy level is very different to the two-ellipses case.
We also will provide a new proof of Poncelet′s result based on dynamical tools.

This is a joint work with Anna Cima and Armengol Gasull.