Non-autonomous curves on surfaces

Michael Khanevsky

doi:10.3934/jmd.2021010

Article Contents

2021, Volume 17: 305-317. Doi: 10.3934/jmd.2021010

This volume Previous Article Direct products, overlapping actions, and critical regularity Next Article On the relation between action and linking

Non-autonomous curves on surfaces

Michael Khanevsky

Mathematics Department, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received: January 28, 2021

Revised: March 2021

Published: June 2021

The author was supported by the Azrieli Fellowship

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

Consider a symplectic surface $ \Sigma $ with two properly embedded Hamiltonian isotopic curves $ L $ and $ L' $. Suppose $ g \in Ham(\Sigma) $ is a Hamiltonian diffeomorphism which sends $ L $ to $ L' $. Which dynamical properties of $ g $ can be detected by the pair $ (L, L') $? We present two scenarios where one can deduce that $ g $ is "chaotic:" non-autonomous or even of positive entropy.

Keywords:

Hamiltonian dynamics,
non-autonomous diffeomorphisms.

Mathematics Subject Classification: Primary: 37J30; Secondary: 37J39.

Citation:

Full Text(HTML)

Figure 1. $ g_{T, \tau} $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	M. Bestvina and K. Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol., 6 (2002), 69-89. doi: 10.2140/gt.2002.6.69.
[2]	M. Brandenbursky and J. Kędra, On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc, Algebr. Geom. Topol., 13 (2012), 795-816. doi: 10.2140/agt.2013.13.795.
[3]	M. Brandenbursky, J. Kędra and E. Shelukhin, On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus, Commun. Contemp. Math., 20 (2018), 27pp. doi: 10.1142/S0219199717500420.
[4]	M. Brandenbursky and M. Marcinkowski, Entropy and quasimorphisms, J. Mod. Dyn., 15 (2019), 143-163. doi: 10.3934/jmd.2019017.
[5]	M. Brandenbursky, Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces, Internat. J. Math., 26 (2015), 29pp. doi: 10.1142/S0129167X15500664.
[6]	D. Calegari, scl, MSJ Memoirs, 20, The Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018.
[7]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[8]	M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012,169–197. doi: 10.1007/978-0-8176-8277-4_8.
[9]	M. Khanevsky, Hofer's length spectrum of symplectic surfaces, J. Mod. Dyn., 9 (2015), 219-235. doi: 10.3934/jmd.2015.9.219.
[10]	L. Polterovich and E. Shelukhin, Autonomous Hamiltonian flows, Hofer's geometry and persistence modules, Selecta Math. (N.S.), 22 (2016), 227-296. doi: 10.1007/s00029-015-0201-2.
[11]	V. Silva, J. Robbin and D. Salamon, Combinatorial Floer homology, Mem. Amer. Math. Soc., 230 (2012).