American Institute of Mathematical Sciences

Advanced
  2020, 16: 289-304. doi: 10.3934/jmd.2020010

Pseudo-rotations and Steenrod squares

Egor Shelukhin

Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada

Received  September 03, 2019 Revised  July 2020 Published  October 2020

In this note we prove that if a closed monotone symplectic manifold $ M $ of dimension $ 2n $, satisfying a homological condition that holds in particular when the minimal Chern number is $ N>n $, admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.

Keywords: Hamiltonian diffeomorphism, pseudo-rotation, quantum Steenrod square, equivariant pair-of-pants product.
Mathematics Subject Classification: Primary: 53D40, 37J10; Secondary: 37J45.
Citation: Egor Shelukhin. Pseudo-rotations and Steenrod squares. Journal of Modern Dynamics, 2020, 16: 289-304. doi: 10.3934/jmd.2020010
References:
[1]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3–36.  Google Scholar

[2]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989.  doi: 10.2140/gt.2009.13.2881.  Google Scholar

[3]

B. Bramham, Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves, Ann. of Math. (2), 181 (2015), 1033-1086.  doi: 10.4007/annals.2015.181.3.4.  Google Scholar

[4]

B. Bramham, Pseudo-rotations with sufficiently Liouvillean rotation number are $C^0$-rigid, Invent. Math., 199 (2015), 561-580.  doi: 10.1007/s00222-014-0525-0.  Google Scholar

[5]

I. Charton, Hamiltonian ${S}^1$-spaces with large equivariant pseudo-index, J. Geom. Phys., 147 (2020), 103521, 10 pp. doi: 10.1016/j.geomphys.2019.103521.  Google Scholar

[6]

E. Cineli and V. L. Ginzburg, On the iterated Hamiltonian Floer homology, preprint, arXiv: 1902.06369, 2019. Google Scholar

[7]

E. Cineli, V. L. Ginzburg and B. Z. Gürel, From pseudo-rotations to holomorphic curves via quantum Steenrod squares, preprint, arXiv: 1909.11967, 2019. Google Scholar

[8]

E. Cineli, V. L. Ginzburg and B. Z. Gürel, Pseudo-rotations and holomorphic curves, preprint, arXiv: 1905.07567, 2019. Google Scholar

[9]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 30 (2003), 1635-1676.  doi: 10.1155/S1073792803210011.  Google Scholar

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520.  doi: 10.1017/S0143385703000798.  Google Scholar

[11]

K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology, 38 (1999), 933-1048.  doi: 10.1016/S0040-9383(98)00042-1.  Google Scholar

[12]

V. L. Ginzburg and B. Z. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics, Geom. Topol., 13 (2009), 2745-2805.  doi: 10.2140/gt.2009.13.2745.  Google Scholar

[13]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[14]

V. L. Ginzburg and B. Z. Gürel, Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms, Duke Math. J., 163 (2014), 565-590.  doi: 10.1215/00127094-2410433.  Google Scholar

[15]

V. L. Ginzburg and B. Z. Gürel, Hamiltonian pseudo-rotations of projective spaces, Invent. Math., 214 (2018), 1081-1130.  doi: 10.1007/s00222-018-0818-9.  Google Scholar

[16]

V. L. Ginzburg and B. Z. Gürel, Conley conjecture revisited, Int. Math. Res. Not. IMRN, 2019 (2019), 761-798.  doi: 10.1093/imrn/rnx137.  Google Scholar

[17]

L. Godinho, F. von Heymann and S. Sabatini, 12, 24 and beyond, Adv. Math., 319 (2017), 472-521.  doi: 10.1016/j.aim.2017.08.023.  Google Scholar

[18]

D. McDuff, Hamiltonian $S^1$-manifolds are uniruled, Duke Math. J., 146 (2009), 449-507.  doi: 10.1215/00127094-2009-003.  Google Scholar

[19]

M. McLean, Local Floer homology and infinitely many simple Reeb orbits, Algebr. Geom. Topol., 12 (2012), 1901-1923.  doi: 10.2140/agt.2012.12.1901.  Google Scholar

[20]

Y.-G. Oh, Symplectic topology and Floer homology. Vols. 1 and 2, New Mathematical Monographs, vol. 27 and 28, Cambridge University Press, Cambridge, 2015.  Google Scholar

[21]

S. Sabatini, On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action, Commun. Contemp. Math., 19 (2017), 1750043, 51pp. doi: 10.1142/S0219199717500432.  Google Scholar

[22]

D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999,143–229.  Google Scholar

[23]

D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.  doi: 10.1002/cpa.3160451004.  Google Scholar

[24]

P. Seidel, The equivariant pair-of-pants product in fixed point Floer cohomology, Geom. Funct. Anal., 25 (2015), 942-1007.  doi: 10.1007/s00039-015-0331-x.  Google Scholar

[25]

P. Seidel and N. Wilkins,, work in progress. Google Scholar

[26]

E. Shelukhin, On the Hofer-Zehnder conjecture, preprint, arXiv: 1905.04769, 2019. Google Scholar

[27]

E. Shelukhin, Pseudo-rotations and Steenrod squares revisited, arXiv: 1909.12315, 2019. Google Scholar

[28]

E. Shelukhin and J. Zhao, The $ {\mathbb{Z}/(p)}$-equivariant product-isomorphism in fixed point Floer homology, preprint, arXiv: 1905.03666, 2019. Google Scholar

[29]

M. Usher and J. Zhang, Persistent homology and Floer–Novikov theory, Geom. Topol., 20 (2016), 3333-3430.  doi: 10.2140/gt.2016.20.3333.  Google Scholar

[30]

N. Wilkins, A construction of the quantum Steenrod squares and their algebraic relations, Geom. Topol., to appear, arXiv: 1810.02738, 2018. Google Scholar

[31]

N. Wilkins, Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology, preprint, arXiv: 1810.02738, 2018. Google Scholar

show all references

References:
[1]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3–36.  Google Scholar

[2]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989.  doi: 10.2140/gt.2009.13.2881.  Google Scholar

[3]

B. Bramham, Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves, Ann. of Math. (2), 181 (2015), 1033-1086.  doi: 10.4007/annals.2015.181.3.4.  Google Scholar

[4]

B. Bramham, Pseudo-rotations with sufficiently Liouvillean rotation number are $C^0$-rigid, Invent. Math., 199 (2015), 561-580.  doi: 10.1007/s00222-014-0525-0.  Google Scholar

[5]

I. Charton, Hamiltonian ${S}^1$-spaces with large equivariant pseudo-index, J. Geom. Phys., 147 (2020), 103521, 10 pp. doi: 10.1016/j.geomphys.2019.103521.  Google Scholar

[6]

E. Cineli and V. L. Ginzburg, On the iterated Hamiltonian Floer homology, preprint, arXiv: 1902.06369, 2019. Google Scholar

[7]

E. Cineli, V. L. Ginzburg and B. Z. Gürel, From pseudo-rotations to holomorphic curves via quantum Steenrod squares, preprint, arXiv: 1909.11967, 2019. Google Scholar

[8]

E. Cineli, V. L. Ginzburg and B. Z. Gürel, Pseudo-rotations and holomorphic curves, preprint, arXiv: 1905.07567, 2019. Google Scholar

[9]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 30 (2003), 1635-1676.  doi: 10.1155/S1073792803210011.  Google Scholar

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520.  doi: 10.1017/S0143385703000798.  Google Scholar

[11]

K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology, 38 (1999), 933-1048.  doi: 10.1016/S0040-9383(98)00042-1.  Google Scholar

[12]

V. L. Ginzburg and B. Z. Gürel, Action and index spectra and periodic orbits in Hamiltonian dynamics, Geom. Topol., 13 (2009), 2745-2805.  doi: 10.2140/gt.2009.13.2745.  Google Scholar

[13]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[14]

V. L. Ginzburg and B. Z. Gürel, Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms, Duke Math. J., 163 (2014), 565-590.  doi: 10.1215/00127094-2410433.  Google Scholar

[15]

V. L. Ginzburg and B. Z. Gürel, Hamiltonian pseudo-rotations of projective spaces, Invent. Math., 214 (2018), 1081-1130.  doi: 10.1007/s00222-018-0818-9.  Google Scholar

[16]

V. L. Ginzburg and B. Z. Gürel, Conley conjecture revisited, Int. Math. Res. Not. IMRN, 2019 (2019), 761-798.  doi: 10.1093/imrn/rnx137.  Google Scholar

[17]

L. Godinho, F. von Heymann and S. Sabatini, 12, 24 and beyond, Adv. Math., 319 (2017), 472-521.  doi: 10.1016/j.aim.2017.08.023.  Google Scholar

[18]

D. McDuff, Hamiltonian $S^1$-manifolds are uniruled, Duke Math. J., 146 (2009), 449-507.  doi: 10.1215/00127094-2009-003.  Google Scholar

[19]

M. McLean, Local Floer homology and infinitely many simple Reeb orbits, Algebr. Geom. Topol., 12 (2012), 1901-1923.  doi: 10.2140/agt.2012.12.1901.  Google Scholar

[20]

Y.-G. Oh, Symplectic topology and Floer homology. Vols. 1 and 2, New Mathematical Monographs, vol. 27 and 28, Cambridge University Press, Cambridge, 2015.  Google Scholar

[21]

S. Sabatini, On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action, Commun. Contemp. Math., 19 (2017), 1750043, 51pp. doi: 10.1142/S0219199717500432.  Google Scholar

[22]

D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999,143–229.  Google Scholar

[23]

D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.  doi: 10.1002/cpa.3160451004.  Google Scholar

[24]

P. Seidel, The equivariant pair-of-pants product in fixed point Floer cohomology, Geom. Funct. Anal., 25 (2015), 942-1007.  doi: 10.1007/s00039-015-0331-x.  Google Scholar

[25]

P. Seidel and N. Wilkins,, work in progress. Google Scholar

[26]

E. Shelukhin, On the Hofer-Zehnder conjecture, preprint, arXiv: 1905.04769, 2019. Google Scholar

[27]

E. Shelukhin, Pseudo-rotations and Steenrod squares revisited, arXiv: 1909.12315, 2019. Google Scholar

[28]

E. Shelukhin and J. Zhao, The $ {\mathbb{Z}/(p)}$-equivariant product-isomorphism in fixed point Floer homology, preprint, arXiv: 1905.03666, 2019. Google Scholar

[29]

M. Usher and J. Zhang, Persistent homology and Floer–Novikov theory, Geom. Topol., 20 (2016), 3333-3430.  doi: 10.2140/gt.2016.20.3333.  Google Scholar

[30]

N. Wilkins, A construction of the quantum Steenrod squares and their algebraic relations, Geom. Topol., to appear, arXiv: 1810.02738, 2018. Google Scholar

[31]

N. Wilkins, Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology, preprint, arXiv: 1810.02738, 2018. Google Scholar

[1]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[2]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
[3]

Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224
[4]

Toshiyuki Suzuki. Semilinear Schrödinger evolution equations with inverse-square and harmonic potentials via pseudo-conformal symmetry. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4347-4377. doi: 10.3934/cpaa.2021163
[5]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719
[6]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063
[7]

Oǧul Esen, Serkan Sütlü. Matched pair analysis of the Vlasov plasma. Journal of Geometric Mechanics, 2021, 13 (2) : 209-246. doi: 10.3934/jgm.2021011
[8]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[9]

Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119
[10]

Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra. Electronic Research Archive, 2020, 28 (2) : 951-959. doi: 10.3934/era.2020050
[11]

Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 867-888. doi: 10.3934/dcds.2018037
[12]

Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017
[13]

Christian Bonatti, Sylvain Crovisier and Amie Wilkinson. The centralizer of a $C^1$-generic diffeomorphism is trivial. Electronic Research Announcements, 2008, 15: 33-43. doi: 10.3934/era.2008.15.33
[14]

Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042
[15]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
[16]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007
[17]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593
[18]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
[19]

Helmut Kröger. From quantum action to quantum chaos. Conference Publications, 2003, 2003 (Special) : 492-500. doi: 10.3934/proc.2003.2003.492
[20]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (180)
  • HTML views (174)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy