
-
Previous Article
On the relation between action and linking
- JMD Home
- This Volume
-
Next Article
Direct products, overlapping actions, and critical regularity
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.
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). |
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). |

[1] |
Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231 |
[2] |
Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743 |
[3] |
Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260 |
[4] |
Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703 |
[5] |
Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108 |
[6] |
Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088 |
[7] |
Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224 |
[8] |
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151 |
[9] |
Hong Lu, Mingji Zhang. Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3553-3576. doi: 10.3934/dcdsb.2020072 |
[10] |
Mirelson M. Freitas, Alberto L. C. Costa, Geraldo M. Araújo. Pullback dynamics of a non-autonomous mixture problem in one dimensional solids with nonlinear damping. Communications on Pure and Applied Analysis, 2020, 19 (2) : 785-809. doi: 10.3934/cpaa.2020037 |
[11] |
Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109 |
[12] |
Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251 |
[13] |
Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084 |
[14] |
Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3745-3765. doi: 10.3934/cpaa.2021129 |
[15] |
Fuzhi Li, Dongmei Xu. Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022087 |
[16] |
Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79 |
[17] |
Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144 |
[18] |
Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351 |
[19] |
Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281 |
[20] |
Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376 |
2020 Impact Factor: 0.848
Tools
Metrics
Other articles
by authors
[Back to Top]