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

2014 ◽  
Vol 35 (6) ◽  
pp. 1681-1722 ◽  
Author(s):  
THIERRY BARBOT ◽  
SÉRGIO R. FENLEY
Keyword(s):  
Homotopy Class ◽  
Closed Orbit ◽  
Dehn Surgery ◽  
Anosov Flow ◽  
Free Homotopy Class ◽  
Graph Manifold ◽  
Finite Power ◽  
Graph Manifolds ◽  
Anosov Flows ◽  
Regular Fibers

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.

scholarly journals Diversified homotopic behavior of closed orbits of some -covered Anosov flows

2014 ◽  
Vol 36 (3) ◽  
pp. 767-780
Author(s):  
SÉRGIO R. FENLEY
Keyword(s):  
Periodic Orbits ◽  
Closed Orbit ◽  
The Other ◽  
Dehn Surgery ◽  
Geodesic Flows ◽  
Closed Orbits ◽  
Anosov Flows

We produce infinitely many examples of Anosov flows in closed $3$-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.

Incompressible tori transverse to Anosov flows in 3-manifolds

1997 ◽  
Vol 17 (1) ◽  
pp. 105-121 ◽  
Author(s):  
SÉRGIO R. FENLEY
Keyword(s):  
Fundamental Group ◽  
Abelian Subgroup ◽  
Closed Orbit ◽  
Anosov Diffeomorphism ◽  
Anosov Flow ◽  
Stable Foliation ◽  
Anosov Flows

We consider Anosov flows in 3-manifolds. Suppose that there is a rank-two free abelian subgroup of the fundamental group of the manifold, so that none of its elements can be represented by a closed orbit of the flow. We then show that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism. As a consequence we prove that if $T$ is an incompressible torus so that no loop in $T$ is freely homotopic to a closed orbit of the flow, then $T$ is isotopic to a transverse torus. Finally, we show that if $T$ is an incompressible torus transverse to the stable foliation, then either there is a closed leaf in the induced foliation in $T$, or the flow is topologically conjugate to a suspension Anosov flow.

scholarly journals Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1330
Author(s):  
Raeyong Kim
Keyword(s):  
Group Structure ◽  
Conjugacy Problem ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Graph Manifold ◽  
Solvable Word Problem ◽  
Algorithmic Problems ◽  
Graph Manifolds ◽  
Dimensional Graph ◽  
Solvable Word

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

scholarly journals Global cross sections for Anosov flows

2015 ◽  
Vol 36 (8) ◽  
pp. 2661-2674
Author(s):  
SLOBODAN N. SIMIĆ
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Anosov Flow ◽  
Anosov Flows ◽  
New Criterion ◽  
Volume Preserving

We provide a new criterion for the existence of a global cross section to a volume-preserving Anosov flow. The criterion is expressed in terms of expansion and contraction rates of the flow and generalizes known results of this type.

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

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

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.

scholarly journals The Asymptotics of the Higher Dimensional Reidemeister Torsion for Exceptional Surgeries Along Twist Knots

2018 ◽  
Vol 61 (1) ◽  
pp. 211-224 ◽  
Author(s):  
Anh T. Tran ◽  
Yoshikazu Yamaguchi
Keyword(s):  
Asymptotic Behavior ◽  
Reidemeister Torsion ◽  
Graph Manifold ◽  
Higher Dimensional ◽  
Graph Manifolds

AbstractWe determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL2()-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coeõcients in the higher dimensional Reidemeister torsion explicitly.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

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.

scholarly journals L-spaces, taut foliations, and graph manifolds

2020 ◽  
Vol 156 (3) ◽  
pp. 604-612 ◽  
Author(s):  
Jonathan Hanselman ◽  
Jacob Rasmussen ◽  
Sarah Dean Rasmussen ◽  
Liam Watson
Keyword(s):  
Graph Manifold ◽  
Graph Manifolds

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

Anosov flows which are uniformly expanding at periodic points

1994 ◽  
Vol 14 (2) ◽  
pp. 299-304 ◽  
Author(s):  
Ursula Hamenstadt
Keyword(s):  
Compact Manifold ◽  
Periodic Points ◽  
Anosov Flow ◽  
Anosov Flows

AbstractA smooth transitive Anosov flow on a compact manifoldNwhich is uniformly (a, b)-expanding at periodic points for 1 <a<bis uniformly (a− ε,b+ ε)-expanding on all of N for all ε > 0.

scholarly journals Approximate solutions of cohomological equations associated with some Anosov flows

1990 ◽  
Vol 10 (2) ◽  
pp. 367-379 ◽  
Author(s):  
Svetlana Katok
Keyword(s):  
Approximate Solutions ◽  
Geodesic Flows ◽  
Curved Surfaces ◽  
Anosov Flow ◽  
3 Dimensional ◽  
Negatively Curved ◽  
Anosov Flows ◽  
Higher Differentiability ◽  
Special Case ◽  
Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

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