## WINTER 2017/18

The talks start at 16.15 in **Seminar Room NA 5/24 **.

**6.3.2017** Arnaud Maret (ETH Zurich) * “Forcing relations for periodic orbits”
*

** Abstract:**

Periodic orbits play a leading role in the study of the dynamics of iterated transformations of a given space. This talk outlines an abstract approach to the analysis of dynamical forcing relations between periodic orbits of a fixed transformation. In other words, we present a strategy to answer the following question: "Under what conditions on a space, does the existence of some periodic orbit imply, in a general way, the existence of other orbits ?". Every periodic orbit is specified by topological properties that will serve as a base for the comparison. The expected forcing relations define a preorder on the set of such specifications. Understanding the preorder is of main interest in this context. The above framework is illustrated at its best by the Sharkovskii Theorem on interval dynamics. In dimension two, achievements have been made towards the establishment of an analogue for surface homeomorphisms.

**26.1.2018** Christoph Thäle (RUB) * *

** Abstract:**

**16.1.2018** Marie-Claude Arnaud (Avignon) *Arnol’d-Liouville theorems in low regularity*

** Abstract:**

Classical Arnol’d-Liouville theorem describes precisely the dynamics of Hamiltonian systems that have enough independent C2 integrals. For such Hamiltonians, it is known that there is an invariant Lagrangian foliation that is symplectically diffeomorphic to the standard one and that the dynamics restricted to every invariant leaf is conjugate to a translation flow. Here we focus on what happens when we have lower regularity. The motivation for studying low regularity is that when a Tonelli Hamiltonian has no conjugate points, only the existence of continuous integrals can be proved.
More precisely, we will raise the question of which continuous Lagrangian foliations are symplectically homeomorphic to the standard one and prove that when the integrals are just C1 and when the Hamiltonian is Tonelli, we indeed obtain a continiuous Lagrangian foliation that is symplectically homeomorphic to the standard one and that Arnol’d-Liouville theorem remains true with a symplectic homeomorphism instead of a C1 change of coordinates.

**19.12.2017** Robert Krawczyk (Gdansk) *Homoclinic orbits for an almost periodically Newtonian system in R^3 *

** Abstract:**

We will be concerned with the existence of homoclinic solutions for a Newtonian system q ̈(t) + a(t)W (q(t)) = 0, where t ∈ R, q ∈ R^3. It is assumed that there is a line l ∈ R^3 \ {0} such that potential W ∈ C2(R^3 \ l,R) has a global maximum at the origin and the line l consists of singular points. Moreover, W satisfies the ”strong-force” condition in a neighbourhood of l and a: R → R is a continuous almost periodic function. The existence of at least two solutions will be discussed.

**12.12.2017** Luis Diogo (Uppsala) *Lifting Lagrangians from Donaldson-type divisors*

** Abstract:**

We prove that there are infinitely many non-symplectomorphic monotone Lagrangian
tori in complex projective spaces, quadrics and cubics of complex dimension at least 3.
This result follows from a relation between the superpotential of a monotone Lagrangian L
in a closed symplectic manifold Y and the superpotential of a Lagrangian lift of L to a
closed symplectic manifold X, in which Y sits as a codimension 2 symplectic submanifold.
This relation sometimes involves relative Gromov-Witten invariants of the pair (X,Y).
The superpotential of a Lagrangian is a count of pseudoholomorphic disks (of Maslov index 2)
with boundary on the Lagrangian, and it plays an important role in Floer theory and mirror
symmetry. This is joint work with D. Tonkonog, R. Vianna and W. Wu.

**12.12.2017** Christian Gloy (Hamburg) * "Overtwisted contact structures in dimension three”*

** Abstract:**

In this talk I will illustrate how M. Borman, Y. Eliashberg and E. Murphy proved in their fa- mous paper from 2014 that there is a parametric extension h-principle for overtwisted contact manifolds (M, ξ) in all dimensions 2n + 1.
Basically, thanks to Gromov’s h-principle for (open) contact manifolds, the whole problem reduces to a local one, more specifically to an embedded annulus C = S2n × [0, 1] inside the manifold M.
After partitioning C into smaller subannuli the main idea is to describe the remaining structure ξ on C by the graphs of sψ,s ∈ [0,1], for a suitable function ψ : S2n → R and achieve the loosely speaking rule: wherever ψ is positive, we can extend the contact structure over C and on regions where ψ could be negative we need to find some special holes of the structure ξ which are modelled by so called (Hamiltonian) contact shells. These shells are 2n + 1-balls BK which are essentially the region under the graph of a smooth function K : ∆ × S1 → R with K|∂∆×S1 > 0, ∆ a cylindrical domain, such that near the boundary of BK the almost contact structure ξ is completely determined by K and actually genuine contact.
At this stage the notion of overtwisted discs and therefore overtwisted structures in higher dimensions comes into play and such discs will be used to fill the remaining special holes wrt. ξ with a genuine structure. It turns out that the structure we end with is indeed an overtwisted one on M.
I will mainly discuss the 3−dimensional case and briefly point out where in higher dimensions more work is involved and the argument becomes more difficult.

**5.12.2017** Felix Schmäschke (Berlin) * On geodesic flows with symmetries and closed magnetic geodesics on orbifolds*

** Abstract:**

This talk is about a joint project with Luca Asselle. In the talk I explain how the magnetic flow on any (effective) orbifold lifts to a geodesic flow on a manifold admitting an action of a compact Lie group. We use this fact to prove that under some topological condition, more precisely that the orbifold is not rationally aspherical, then there exists a closed magnetic geodesic for almost every energy. We also give an account on how our approach yields an alternative proof of the known theorem by Guruprasad and Haeflinger about the existence of closed geodesics on orbifolds.

**28.11.2017** Stefano Luzzatto (ICTP, Trieste) *"Physical measures for dynamical systems"
*

** Abstract:**

We present the Palis and Viana conjectures on the existence of physical measures, review some of the literature, and describe some recent joint work with Climenhaga and Pesin on the existence of Sinai-Ruelle-Bowen physical measures for nonuniformly hyperbolic surface diffeomorphisms.

**28.11.2017** Daniel Rosen (Tel Aviv) *"Dual caustics in Minkowski billiards"*

** Abstract:**

Mathematical billiards are a classical and well-studied class of dynamical systems, "a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards no non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have
equal perimeters and other classical parameters. We will show that, when the norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails.
Based on joint work with S. Artstein-Avidan, D. Florentin, and Y. Ostrover.

**21.11.2017** Marco Mazzucchelli (ENS Lyon) * “On the boundary rigidity problem for surfaces”
*

** Abstract:**

The classical boundary rigidity problem asks whether, or to what extent, the inner geometry of a compact Riemannian manifold with boundary can be determined by means of boundary measurements, such as the distance function among boundary points, or the geodesic scattering map. In my talk I will review this problem and some of the known results that are valid for "simple" Riemannian manifolds. I will then sketch the proof of some recent boundary rigidity results for non-simple Riemannian surfaces, including surfaces with trapped geodesics or with non-convex boundary. The talk is based on joint work with Colin Guillarmou and Leo Tzou.

**7.11.2017** Seongchan Kim (Augsburg) * J^+-like invariants and the restricted three body problem*

** Abstract:**

Recently, Cieliebak-Frauenfelder-van Koert observed disasters which can happen in a family of periodic orbits in the planar circular restricted three body problem and defined new invariants of such families, based on Arnold's $J^+$-invariant. In this talk, we recall their results and determine those invariants for a distinguished class of periodic orbits in the restricted three body problem, i.e., periodic orbits of the second kind. This talk is based on a joint work with Joontae Kim.

**24.10.2017** Carsten Haug (Neuchâtel) * “On action selectors”
*

** Abstract:**

An action selector associates to every Hamiltonian function the
action of one of its periodic orbits, in a continuous way.
The mere existence of an action selector has many consequences
in symplectic dynamics and geometry (like Gromov's non-squeezing
theorem and the existence of closed orbits on energy surfaces of contact type).
The first selectors were constructed for the standard symplectic vector space R^2n
by Viterbo and Hofer-Zehnder, and then for (essentially) all symplectic manifolds by
means of Floer homology (Schwarz, Oh, Usher).
I will describe a more elementary construction of an action selector for manifolds $(M,\omega)$
with $[\omega] | \pi_2(M) = 0$, that uses only Gromov compactness.
This is joint work with Alberto Abbondandolo and Felix Schlenk.