Abstract
We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed $2$-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular, every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.
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author = {B{é}guin, F. and Crovisier, S. and Le Roux, F. and Patou, A.},
title = {Pseudo-rotations of the closed annulus: variation on a theorem of {J}. {K}wapisz},
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author = {Hofer, Helmut},
title = {Holomorphic curves and dynamics in dimension three},
booktitle = {Symplectic Geometry and Topology},
venue = {{P}ark {C}ity, {UT},
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address = {Boston},
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zblnumber = {0924.58003},
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mrreviewer = {Kai Cieliebak},
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} -
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@incollection{HWZ_survey, mrkey = {1928532},
author = {Hofer, Helmut and Wysocki, K. and Zehnder, E.},
title = {Pseudoholomorphic curves and dynamics in three dimensions},
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pages = {1129--1188},
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doi = {10.1016/S1874-575X(02)80017-0},
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address = {Zürich},
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doi = {10.2307/1971131},
} -
[Shnirelman] L. G. Shnirelman, "An example of a transformation of the plane (Russian)," An example of a transformation of the plane (Russian), vol. 14, pp. 64-74, 1930.
@article{Shnirelman,
author = {Shnirelman, L. G.},
title = {An example of a transformation of the plane ({R}ussian)},
journal = {An example of a transformation of the plane (Russian)},
volume = {14},
year = {1930},
pages = {64--74},
note = {(Science section, Fis-math. part)},
} -
[S_intersections]
R. Siefring, "Intersection theory of punctured pseudoholomorphic curves," Geom. Topol., vol. 15, iss. 4, pp. 2351-2457, 2011.
@article{S_intersections, mrkey = {2862160},
author = {Siefring, Richard},
title = {Intersection theory of punctured pseudoholomorphic curves},
journal = {Geom. Topol.},
fjournal = {Geometry \& Topology},
volume = {15},
year = {2011},
number = {4},
pages = {2351--2457},
issn = {1465-3060},
mrclass = {53D42 (32Q65 53D35 53D45)},
mrnumber = {2862160},
mrreviewer = {Jonathan David Evans},
doi = {10.2140/gt.2011.15.2351},
} -
[Wendl_thesis]
C. Wendl, Finite Energy Foliations and Surgery on Transverse Links, Ann Arbor, MI: ProQuest LLC, 2005.
@book{Wendl_thesis, mrkey = {2706831},
author = {Wendl, Chris},
title = {Finite Energy Foliations and Surgery on Transverse Links},
note = {Thesis (Ph.D.)--New York University},
publisher = {ProQuest LLC},
address = {Ann Arbor, MI},
year = {2005},
pages = {411},
isbn = {978-0496-90328-3},
mrclass = {Thesis},
mrnumber = {2706831},
url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3157866},
} -
[Wendl_compactness]
C. Wendl, "Compactness for embedded pseudoholomorphic curves in 3-manifolds," J. Eur. Math. Soc. $($JEMS$)$, vol. 12, iss. 2, pp. 313-342, 2010.
@article{Wendl_compactness, mrkey = {2608942},
author = {Wendl, Chris},
title = {Compactness for embedded pseudoholomorphic curves in 3-manifolds},
journal = {J. Eur. Math. Soc. $($JEMS$)$},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {12},
year = {2010},
number = {2},
pages = {313--342},
issn = {1435-9855},
mrclass = {53D45 (53D35 57R17)},
mrnumber = {2608942},
mrreviewer = {Timothy Perutz},
doi = {10.4171/JEMS/199},
} -
[Wendl_lef_fib]
C. Wendl, "Strongly fillable contact manifolds and $J$-holomorphic foliations," Duke Math. J., vol. 151, iss. 3, pp. 337-384, 2010.
@article{Wendl_lef_fib, mrkey = {2605865},
author = {Wendl, Chris},
title = {Strongly fillable contact manifolds and {$J$}-holomorphic foliations},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {151},
year = {2010},
number = {3},
pages = {337--384},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {53D35 (53D45 57R17)},
mrnumber = {2605865},
mrreviewer = {Tobias Ekholm},
doi = {10.1215/00127094-2010-001},
} -
[Wendl_boundary] C. Wendl, Punctured holomorphic curves with boundary in 3-manifolds: Fredholm theory and embededdness.
@misc{Wendl_boundary,
author = {Wendl, Chris},
title = {Punctured holomorphic curves with boundary in 3-manifolds: {F}redholm theory and embededdness},
note = {in preparation},
}