On curves with the Poritsky property

@article{Glutsyuk2019OnCW,
title={On curves with the Poritsky property},
author={A. A. Glutsyuk},
journal={Journal of Fixed Point Theory and Applications},
year={2019},
volume={24},
pages={1-60}
}
• A. Glutsyuk
• Published 7 January 2019
• Mathematics
• Journal of Fixed Point Theory and Applications
Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve $$\gamma$$ γ , the string construction yields a one-parameter family $$\Gamma _p$$ Γ p of nested billiard tables containing $$\gamma$$ γ for which $$\gamma$$ γ is a caustic: the reflection from $$\Gamma _p$$ Γ p sends each tangent line to $$\gamma$$ γ to a line tangent to $$\gamma$$ γ . The reflections from $$\Gamma _p$$ Γ p act on the corresponding tangency points, inducing a family of string…
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By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A
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The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a local
To a closed convex smooth curve in the plane the dual billiard transformation of its exterior corresponds: given a point outside of the curve, draw a tangent line to it through the point, and reflect
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The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and