Given two ellipses, one surrounding the other one, Poncelet introduced 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. We recall what he proved around 1814 in the
dynamical systems language: In the two ellipses case and when the rotation number of
P is rational there exists a n ∈ N such that Pn = Id, or in other words, the Poncelet’s
map is conjugate to a rational rotation. In this paper 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 conjugate to a rotation. Finally, we also
provide a new proof of Poncelet’s result based on dynamical and computational tools.
CitationCima, A.; Gasull, A.; Mañosa, V. On Poncelet's maps. "Computers and mathematics with applications", 08 Agost 2010, vol. 60, núm. 5, p. 1457-1464.
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