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2021, 17: 305-317.
doi: 10.3934/jmd.2021010
Non-autonomous curves on surfaces
Mathematics Department, Technion-Israel Institute of Technology, Haifa, 32000, Israel |
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:
Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics,
2021, 17: 305-317.
doi: 10.3934/jmd.2021010
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M. Khanevsky,
Hofer's length spectrum of symplectic surfaces, J. Mod. Dyn., 9 (2015), 219-235.
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L. Polterovich and E. Shelukhin,
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V. Silva, J. Robbin and D. Salamon, Combinatorial Floer homology, Mem. Amer. Math. Soc., 230 (2012). |
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. |
| [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). |
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