The main topic of this event is the dynamical study of mathematical billiards.
Inverse problems and rigidity questions in Billiard Dynamics (Alfonso Sorrentino)
|A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain (e.g. the shape of the billiard table): while it is evident how the shape determines the dynamics, a more subtle and difficult question is to which extent the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing unanswered questions and difficult conjectures that have been the focus of very active research over the last decades. In this talk I shall describe several of these questions, with particular emphasis on recent re- sults related to the classification of integrable billiards (also known as Birkhoff conjecture), and to the possibility of inferring dynamical information on the billiard map from its Length Spectrum (i.e., the lengths of its periodic orbits). This talk is based on joint works with Guan Huang and Vadim Kaloshin.|
Generic dynamics of multidimensional billiards (José Pedro Gaivão)
In this talk we discuss generic properties of billiards inside strictly smooth convex bodies. In particular, we show that in a C2-open and dense set of strictly convex bodies, the associated multidimensional billiard maps have positive topological entropy.
Chaotic motion in the breathing circle billiard (Stefano Marò)
|We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on Aubry-Mather theory.|
Polygonal billiards with contracting reflection laws (Gianluigi Del Magno)
|The dynamics of billiards has been studied in great detail when the reflection law is the specular one. This presentation regards polygonal billiards with contracting reflection laws, i.e. non-specular reflection laws such that the function specifying the dependence of the angle of reflection on the angle of incidence is a contraction. The dynamics of such billiards differs greatly from that of polygonal billiards with the specular reflection law: whereas the latter are never hyperbolic systems, generic polygonal billiards with a contracting reflection law exhibit uniformly hyperbolic attractors supporting a finite number of ergodic physical measures. In this presentation, I will survey the main results obtained for this new class of billiards jointly with P. Duarte, J. Lopes Dias, J. P. Gaivão and D. Pinheiro.|
|Time (GMT+1)||July 19|
|15:30-16:00||José Pedro Gaivão|
|16:30-17:00||Gianluigi Del Magno|
This event is addressed to researchers and advanced students in mathematics.
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