Growth of maps, distortion in groups and symplectic geometry

Discontinuity growth of interval exchange maps

For an interval exchange map, the number of discontinuities of its iterates either exhibits linear growth or is bounded. This dichotomy is used to prove that the group of interval exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval exchanges. As a further application of this dichotomy, a classification of centralizers in the group of interval exchanges is given. This classification of centralizers is used to compute the automorphism group of the interval exchange group.

On primes and period growth for Hamiltonian diffeomorphisms

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we partially recover, using only symplectic too…

关于星形超曲面上Reeb轨道增长速率的研究

On the growth rate of Reeb orbit on star-shaped hypersurfaces

研究星形超曲面上的Reeb轨道增长速率，证明在特定拓扑条件下，存在无限多个简单闭轨，且其数量随时间增长至少与素数分布一致。

Fiberwise volume growth via Lagrangian intersections

We consider Hamiltonian diffeomorphisms $ϕ$ of the unit cotangent bundle over a closed Riemannian manifold $(M,g)$ which extend to Hamiltonian diffeomorphisms of $T^*M$ equal to the time-1-map of the geodesic flow for $|p| \ge 1$. For such diffeomorphisms we establish uniform lower bounds for the fiberwise volume growth of $ϕ$ which were previously known for geodesic flows and which depend only on $(M,g)$ or on the homotopy type of $M$. More precisely, we show that for each $q \in M$ the volume growth of the unit ball in $T_q^*M$ under the iterates of $ϕ$ is at least linear if $M$ is rationall…

Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps

We consider the learning and prediction of nonlinear time series generated by a latent symplectic map. A special case is (not necessarily separable) Hamiltonian systems, whose solution flows give such symplectic maps. For this special case, both generic approaches based on learning the vector field of the latent ODE and specialized approaches based on learning the Hamiltonian that generates the vector field exist. Our method, however, is different as it does not rely on the vector field nor assume its existence; instead, it directly learns the symplectic evolution map in discrete time. Moreove…

共形辛系统、法向双曲不变流形与散射映射的几何、拓扑和动力学性质

Geometric, topological and dynamical properties of conformally symplectic systems, normally hyperbolic invariant manifolds, and scattering maps

本文研究共形辛微分同胚中流形拓扑、映射动力学与不变流形几何的深层联系，证明NHIM的辛性等价于特定增长率配对规则与不等式，并证明散射映射在耗散动力学下仍保持辛性，同时将结果推广到预辛形式情形。