Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

Typical representatives of free homotopy classes in multi-punctured plane

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

Marked length rigidity for one-dimensional spaces

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function [Formula: see text] (the value [Formula: see text] being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset [Formula: see text], which is the union of all non-constant minimal loops of finite length. We show that if [Formula: see text] is a compact, non-contractible, geodesic space of topological dimension one, then [Formula: see text] deformation retracts to [Formula: see text]. Moreover, [Formula: see text] can be characterized as the minimal subset of [Formula: see text] to which [Formula: see text] deformation retracts. Let [Formula: see text] be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set [Formula: see text]. We prove that any isomorphism [Formula: see text] satisfying [Formula: see text], forces the existence of an isometry [Formula: see text] which induces the map [Formula: see text] on the level of fundamental groups. Thus, for compact, non-contractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset [Formula: see text] up to isometry.

On the filtered symplectic homology of prequantization bundles

We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit — the linking number filtration — and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e. the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

Smooth and Topological Almost Concordance

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Y × [0, 1] and the action of the concordance group of knots in S3 that ties in local knots. We prove that the trivial free homotopy class in every 3-manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not smoothly almost concordant to one another. Then, in every lens space and for every free homotopy class, we find a pair of topologically concordant but not smoothly almost concordant knots. Finally, as a topological counterpoint to these results, we show that in every lens space every free homotopy class contains infinitely many topological almost concordance classes.

A generalization of Turaev’s virtual string cobracket and self-intersections of virtual strings

Previously we defined an operation [Formula: see text] that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that [Formula: see text] gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that [Formula: see text] gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings [Formula: see text] such that [Formula: see text] gives a formula for the minimal self-intersection number [Formula: see text]. Finally, we compare the bound given by [Formula: see text] to a bound given by Turaev’s based matrix invariant [Formula: see text], and construct an example that shows the bound on the minimal self-intersection number given by [Formula: see text] is sometimes stronger than the bound [Formula: see text].

A combinatorial algorithm for visualizing representatives with minimal self-intersection

Given an orientable surface with boundary and a free homotopy class of a closed curve on this surface, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self-intersection.

Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds.Geom. Topol. 17(2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of$M$mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.