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Register a new userLens Rigidity in Scattering by Unions of Strictly Convex Bodies in $R^2$
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It was proved in cite{NS1} that obstacles $K$ in $R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
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Obstacles $K$ and $L$ in $R^d$ ($dgeq 2$) are considered that are finite disjoint unions of strictly convex domains with $C^3$ boundaries. We show that if $K$ and $L$ have (almost) the same scattering length spectrum, or (almost) the same traveling times, then $K = L$.
We prove that if two non-trapping obstacles in $mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
Let $N$ be a compact manifold with a foliation $mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,mathscr{F}_N) rightarrow (M,mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,mathscr{F}_N)$ and $h(M,mathscr{F}_M)$ and it holds $h(M,mathscr{F}_M) leq h(N,mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.
For a compact convex set F in R^n, with the origin in its interior, we present a formula to compute the curvature at a fixed point on its boundary, in the direction of any tangent vector. This formula is equivalent to the existing ones, but it is easier to apply.
In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.
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