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

Non-autonomous curves on surfaces

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

Received  January 28, 2021 Revised  March 2021 Published  June 2021

Fund Project: The author was supported by the Azrieli Fellowship

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.

Citation: Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010
References:
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show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Brandenbursky and M. Marcinkowski, Entropy and quasimorphisms, J. Mod. Dyn., 15 (2019), 143-163.  doi: 10.3934/jmd.2019017.  Google Scholar

[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.  Google Scholar

[6]

D. Calegari, scl, MSJ Memoirs, 20, The Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018.  Google Scholar

[7]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676.  doi: 10.1155/S1073792803210011.  Google Scholar

[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.  Google Scholar

[9]

M. Khanevsky, Hofer's length spectrum of symplectic surfaces, J. Mod. Dyn., 9 (2015), 219-235.  doi: 10.3934/jmd.2015.9.219.  Google Scholar

[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.  Google Scholar

[11]

V. Silva, J. Robbin and D. Salamon, Combinatorial Floer homology, Mem. Amer. Math. Soc., 230 (2012).  Google Scholar

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