An extended Poncelet problem I

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An algebraic oval is a smooth compact connected component of a real plane algebraic curve, homeomorphic to a circle. Choose convex ovals $\gamma,\Gamma$ with $\gamma$ strictly inside the region bounded by $\Gamma$. The Poncelet map $P:\Gamma\to\Gamma$ follows a tangent line to the inner oval from $p\in\Gamma$ to its other intersection with $\Gamma$, consistently choosing the first of the two tangent intersections encountered in the counterclockwise direction. Only pairs for which this procedure defines a circle homeomorphism are considered. Rotation conjugacy means there are a homeomorphism $h:\Gamma\to\mathbb R/\mathbb Z$ and $\rho\in\mathbb R/\mathbb Z$ with $h(P(p))=h(p)+\rho$ for all $p$; no rationality condition on $\rho$ is imposed. A curve of degree $d$ is given by an irreducible real polynomial of total degree $d$. The chosen oval is part of that curve; the curve may have other real components.

Statement recorded in the supplied source.

Are there two such irreducible curves of degrees $n,m$ with $n+m>4$, each carrying a nested convex oval as above, whose well-defined Poncelet map is conjugate to a circle rotation? [S2]

Short English statement

Can nested ovals of irreducible curves beyond the two-conic case have a rotation-conjugate Poncelet map?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4700023 (AMR-046-0023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Gasull, Some open problems in low dimensional dynamical systems (2020), Section 6.2, definition of the Poncelet map, Theorem 6.2 and full Problem 23, pp.23--24. https://arxiv.org/pdf/2012.02524.