American Institute of Mathematical Sciences

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  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

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

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