Dynamical spectral rigidity and determination for billiard systems

反问题的目标是从一组粗略的观测结果中重建一个对象。逆问题构成了一个非常广泛的问题类别，几乎在任何科学领域都有深远的应用：高能物理学（识别散射事件中产生的粒子），医学（CT扫描），天体物理学（探测距离我们数十亿光年的恒星光球中的元素），图像处理（数字图像的去噪和去模糊），大数据和机器学习。在这个提议中，我将描述一个可以在（经典）动力系统的背景下纯粹公式化的反问题的分析。考虑一个粒子在平面区域内自由运动的轨迹，并且在与区域边界碰撞时受到弹性反射。我们把那些在有限时间内重复自己的轨迹称为周期性轨迹：这样的轨迹沿着域内的封闭多边形。我们把所有这样的多边形的周长集合称为定义域的长度谱。然后，我们可以制定以下逆动力学问题：动态谱确定：是否有可能通过其长度谱的知识来识别域（模等距）？上述问题，例如在非常自然的光滑域类别中，仍然是一个广泛开放的问题，并且无可争议地被认为是一个极具挑战性的问题。Sarnak推测，光滑域是由它们的拉普拉斯谱局部决定的（这个问题——引用M. Kac的话——经常被表述为：“人能听到鼓的形状吗？”）。由于由波迹公式建立的拉普拉斯（量子）和动力学（经典）谱问题之间的紧密联系，我们发现在动力学环境下研究这一猜想是很自然的。作为第一步，我们可以考虑一个变形问题：我们说一个域是动态光谱刚性的，如果该域的所有光滑变形都必须保持其长度谱是等距的。在过去的几年里，我和我的合作者一起，证明了足够靠近圆盘的对称凸台球的动态谱刚性。此外，我们证明了一类（对称）解析开色散台球的动态谱确定（这些系统的动力学使人想起负曲率流形上的测地线流）。在接下来的几年里，我和我的研究团队将利用为上述结果开发的突破性技术，朝着Sarnak猜想的方向发展。一方面，我将着手证明光滑凸台球（可能具有对称性）的（局部）谱确定结果；另一方面，我将把我在双曲台球上的工作推向光滑范畴的（局部）谱确定问题。我相信，这些成果将在本十年实现，探索基金将在其发展中发挥重要作用。

The goal of inverse problems is the reconstruction of an object from a set of coarse observations. Inverse problems constitute a surprisingly broad class of problems with far-reaching applications in virtually any field of science: high energy physics (identifying particles created in scattering events), medicine (CT scans), astrophysics (detecting elements in the photosphere of stars which are billions of light-years away from us), image processing (de-noising and de-blurring of digital images), Big Data and Machine Learning. In this proposal I will describe the analysis of an inverse problem that can be purely formulated in the context of (classical) dynamical systems. Consider the trajectories of a particle that moves freely inside a planar domain and is subject to elastic reflections upon collisions with the boundary of the domain. We call periodic those trajectories that repeat themselves after a finite amount of time: such trajectories trace closed polygons inscribed in the domain. We call Length Spectrum of the domain the set of perimeters of all such polygons. We can then formulate the following inverse dynamical problem: Dynamical Spectral Determination: is it possible to identify the domain (modulo isometries) by the knowledge of its Length Spectrum? The above question, e.g. in the very natural class of smooth domains, is still wide open, and it is unarguably considered an extremely challenging problem. Sarnak conjectured that smooth domains are locally determined by their Laplace spectrum (a question that -quoting M. Kac- is often phrased as: "Can one hear the shape of a drum?"). Due to the tight connection between the Laplace (quantum) and dynamical (classical) spectral problems established by the Wave Trace formula, we find natural to study this conjecture in the dynamical setting. As a first step, we may consider a deformational problem: we say that a domain is dynamically spectrally rigid if all smooth deformations of the domain that preserve its Length Spectrum are necessarily isometries. In the past few years, together with my collaborators, we proved dynamical spectral rigidity for symmetric convex billiards close enough to disks. Also, we proved dynamical spectral determination for a class of (symmetric) analytic open dispersing billiards (such are systems whose dynamics is reminiscent of geodesic flow on manifolds with negative curvature). In the next several years, my research team and I will leverage on the breakthrough techniques that have been developed for the above results to move towards Sarnak's conjecture. On the one hand I will set to prove (local) spectral determination results for smooth convex billiards (possibly with symmetries); on the other hand I will push my work on hyperbolic billiards towards the problem of (local) spectral determination in the smooth category. I believe that these results will be attainable in this decade, and the Discovery Grant will play a major role in their development.