TY - JOUR AB - The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics. AU - Huang, Guan AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8422 IS - 2 JF - Geometric and Functional Analysis KW - Geometry and Topology KW - Analysis SN - 1016-443X TI - Nearly circular domains which are integrable close to the boundary are ellipses VL - 28 ER - TY - JOUR AB - 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 complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains. AU - Kaloshin, Vadim AU - Sorrentino, Alfonso ID - 8421 IS - 1 JF - Annals of Mathematics KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability SN - 0003-486X TI - On the local Birkhoff conjecture for convex billiards VL - 188 ER - TY - JOUR AB - We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case. AU - Kaloshin, Vadim AU - Zhang, Ke ID - 8420 IS - 11 JF - Nonlinearity KW - Mathematical Physics KW - General Physics and Astronomy KW - Applied Mathematics KW - Statistical and Nonlinear Physics SN - 0951-7715 TI - Density of convex billiards with rational caustics VL - 31 ER - TY - JOUR AB - For any strictly convex planar domain Ω ⊂ R2 with a C∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and Ω¯ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sn}n≥1 (resp. {S¯n}n⩾1) of period going to infinity such that Sn and S¯n have the same period and perimeter for each n. AU - Buhovsky, Lev AU - Kaloshin, Vadim ID - 8426 JF - Regular and Chaotic Dynamics SN - 1560-3547 TI - Nonisometric domains with the same Marvizi-Melrose invariants VL - 23 ER - TY - JOUR AB - The development of strategies to assemble microscopic machines from dissipative building blocks are essential on the route to novel active materials. We recently demonstrated the hierarchical self-assembly of phoretic microswimmers into self-spinning microgears and their synchronization by diffusiophoretic interactions [Aubret et al., Nat. Phys., 2018]. In this paper, we adopt a pedagogical approach and expose our strategy to control self-assembly and build machines using phoretic phenomena. We notably introduce Highly Inclined Laminated Optical sheets microscopy (HILO) to image and characterize anisotropic and dynamic diffusiophoretic interactions, which cannot be performed by conventional fluorescence microscopy. The dynamics of a (haematite) photocatalytic material immersed in (hydrogen peroxide) fuel under various illumination patterns is first described and quantitatively rationalized by a model of diffusiophoresis, the migration of a colloidal particle in a concentration gradient. It is further exploited to design phototactic microswimmers that direct towards the high intensity of light, as a result of the reorientation of the haematite in a light gradient. We finally show the assembly of self-spinning microgears from colloidal microswimmers and carefully characterize the interactions using HILO techniques. The results are compared with analytical and numerical predictions and agree quantitatively, stressing the important role played by concentration gradients induced by chemical activity to control and design interactions. Because the approach described hereby is generic, this works paves the way for the rational design of machines by controlling phoretic phenomena. AU - Aubret, Antoine AU - Palacci, Jérémie A ID - 9053 IS - 47 JF - Soft Matter KW - General Chemistry KW - Condensed Matter Physics SN - 1744-683X TI - Diffusiophoretic design of self-spinning microgears from colloidal microswimmers VL - 14 ER -