forcing theory
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2021 ◽  
pp. 1-44
Author(s):  
MARCELO R. R. ALVES ◽  
ABROR PIRNAPASOV

Abstract We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.



2017 ◽  
Vol 212 (2) ◽  
pp. 619-729 ◽  
Author(s):  
P. Le Calvez ◽  
F. A. Tal


Axiomathes ◽  
2015 ◽  
Vol 25 (4) ◽  
pp. 433-455 ◽  
Author(s):  
Stathis Livadas




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