An extended Poncelet problem II

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. Let $\gamma=\{(x,y):x^2+y^2=1\}$. Choose real polynomials $p_2,p_m$ with $\deg p_2=2$ and $\deg p_m=m\ge3$, such that $p_2=0$ is an ellipse surrounding the unit circle. For sufficiently small real $\varepsilon$, let $\Gamma_\varepsilon$ be the smooth oval near that ellipse in $p_2+\varepsilon p_m=0$. Consider parameters for which this polynomial is irreducible, the oval remains convex and surrounds $\gamma$, and the Poncelet map is well defined.

Statement recorded in the supplied source.

For every such perturbation, is $P_\varepsilon$ conjugate to a circle rotation if and only if $\varepsilon=0$, throughout a sufficiently small neighbourhood of zero? [S2]

Short English statement

Does every irreducible higher-degree perturbation of the outer ellipse destroy rotation conjugacy, however small the perturbation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4700024 (AMR-046-0024). 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 and full Problem 24, pp.23--24. https://arxiv.org/pdf/2012.02524.