September  2018, 38(9): 4571-4602. doi: 10.3934/dcds.2018200

Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms

College of Mathematics, Sichuan University, Chengdu 610065, China

Received  November 2017 Revised  April 2018 Published  June 2018

Fund Project: The author is supported by the Fundamental Research Funds for the central Universities No.YJ201660

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal isotopy such that we can give a fine description of the dynamics of its transverse foliation. We will define a sort of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $ρ(I, z)$ (which is well defined when one blows up $f$ at $z$) is contained in $(-1, 1)$. We will prove the existence of torsion-low maximal isotopies, and will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.

Citation: Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4571-4602. doi: 10.3934/dcds.2018200
References:
[1]

F. Béguin, S. Crovisier and F. Le Roux, Fixed point sets of isotopies on surfaces, preprint, arXiv: 1610.00686v2. Google Scholar

[2]

P. Dávalos, On torus homeomorphisms whose rotation set is an interval, Math. Z., 275 (2013), 1005-1045.  doi: 10.1007/s00209-013-1168-3.  Google Scholar

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P. Dávalos, On the annular maps of the torus and sublinear diffusion, Journal of the Institute of Mathematics of Jussieu, (Published online 2016). doi: 10.1017/S1474748016000268.  Google Scholar

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J. M. GambaudoP. Le Calvez and É. Pécou, Une généralisation d'un théorème de Naishul, (French)[A generalization of a theorem of Naishul], C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 397-402.   Google Scholar

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O. Jaulent, Existence d'un feuilletage positivement transverse á un homéomorphisme de surface, Ann. Inst. Fourier, 64 (2014), 1441-1476.  doi: 10.5802/aif.2886.  Google Scholar

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A. KoropeckiP. Le Calvez and M. Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J., 164 (2015), 403-472.  doi: 10.1215/00127094-2861386.  Google Scholar

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A. Koropecki and F. A. Tal, Bounded and unbounded behavior for area-preserving rational pseudo-rotations, Proc. London Math. Soc., 109 (2014), 785-822.  doi: 10.1112/plms/pdu023.  Google Scholar

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A. Koropecki and F. A. Tal, Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.  doi: 10.1007/s00222-013-0470-3.  Google Scholar

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P. Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, (French) [An equivariant foliated version of Brouwer's translation theorem], Publ. Math. Inst. Hautes Études Sci., 102 (2005), 1-98.  doi: 10.1007/s10240-005-0034-1.  Google Scholar

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P. Le Calvez, Periodic orbits of Hamiltonian homeomorphisms of surfaces, Duke Math. J., 133 (2006), 125-184.  doi: 10.1215/S0012-7094-06-13315-X.  Google Scholar

[12]

P. Le Calvez, Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?, (French) [Why do the periodic points of plane homomorphisms rotate around certain fixed points?], Ann. Sci. Éc. Norm. Supér., 41 (2008), 141-176.  doi: 10.24033/asens.2065.  Google Scholar

[13]

P. Le Calvez and F. A. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, Invent. Math., 212 (2018), 619-729.  doi: 10.1007/s00222-017-0773-x.  Google Scholar

[14]

F. Le Roux, A topological characterization of holomorphic parabolic germs in the plane, Fund. Math., 198 (2008), 77-94.  doi: 10.4064/fm198-1-4.  Google Scholar

[15]

F. Le Roux, L'ensemble de rotation autour d'un point fixe, (French) [The rotation set around a fixed point for surface homeomorphisms], Astérisque 350 (2013), x+109pp.  Google Scholar

[16]

S. Matsumoto, Types of fixed points of index one of surface homeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 1181-1211.  doi: 10.1017/S0143385701001559.  Google Scholar

[17]

S. Matsumoto, Prime end rotation numbers of invariant seperating continua of annulus homeomorphisms, Proc Am. Math. Soc., 140 (2012), 839-845.  doi: 10.1090/S0002-9939-2011-11435-7.  Google Scholar

[18]

G. S. McCarty, Homeotopy groups, Trans. Amer. Math. Soc., 106 (1963), 293-304.  doi: 10.1090/S0002-9947-1963-0145531-9.  Google Scholar

[19]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. doi: 10.1093/oso/9780198794899.001.0001.  Google Scholar

[20]

V. A. Naǐshul', Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in $\textbf{C}^{2}$ and $\textbf{C}P^{2}$, Trudy Moskov. Mat. Obshch., 44 (1982), 235-245.   Google Scholar

[21]

J. Yan, Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms, Ergodic Theory Dynam. Systems, 36 (2016), 2293-2333.  doi: 10.1017/etds.2015.18.  Google Scholar

show all references

References:
[1]

F. Béguin, S. Crovisier and F. Le Roux, Fixed point sets of isotopies on surfaces, preprint, arXiv: 1610.00686v2. Google Scholar

[2]

P. Dávalos, On torus homeomorphisms whose rotation set is an interval, Math. Z., 275 (2013), 1005-1045.  doi: 10.1007/s00209-013-1168-3.  Google Scholar

[3]

P. Dávalos, On the annular maps of the torus and sublinear diffusion, Journal of the Institute of Mathematics of Jussieu, (Published online 2016). doi: 10.1017/S1474748016000268.  Google Scholar

[4]

J. M. GambaudoP. Le Calvez and É. Pécou, Une généralisation d'un théorème de Naishul, (French)[A generalization of a theorem of Naishul], C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 397-402.   Google Scholar

[5]

M. E. Hamstrom, Homotopy groups of the space of homeomorphisms on a 2-manifold, Illinois J. Math., 10 (1966), 563-573.   Google Scholar

[6]

O. Jaulent, Existence d'un feuilletage positivement transverse á un homéomorphisme de surface, Ann. Inst. Fourier, 64 (2014), 1441-1476.  doi: 10.5802/aif.2886.  Google Scholar

[7]

A. KoropeckiP. Le Calvez and M. Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J., 164 (2015), 403-472.  doi: 10.1215/00127094-2861386.  Google Scholar

[8]

A. Koropecki and F. A. Tal, Bounded and unbounded behavior for area-preserving rational pseudo-rotations, Proc. London Math. Soc., 109 (2014), 785-822.  doi: 10.1112/plms/pdu023.  Google Scholar

[9]

A. Koropecki and F. A. Tal, Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.  doi: 10.1007/s00222-013-0470-3.  Google Scholar

[10]

P. Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, (French) [An equivariant foliated version of Brouwer's translation theorem], Publ. Math. Inst. Hautes Études Sci., 102 (2005), 1-98.  doi: 10.1007/s10240-005-0034-1.  Google Scholar

[11]

P. Le Calvez, Periodic orbits of Hamiltonian homeomorphisms of surfaces, Duke Math. J., 133 (2006), 125-184.  doi: 10.1215/S0012-7094-06-13315-X.  Google Scholar

[12]

P. Le Calvez, Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?, (French) [Why do the periodic points of plane homomorphisms rotate around certain fixed points?], Ann. Sci. Éc. Norm. Supér., 41 (2008), 141-176.  doi: 10.24033/asens.2065.  Google Scholar

[13]

P. Le Calvez and F. A. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, Invent. Math., 212 (2018), 619-729.  doi: 10.1007/s00222-017-0773-x.  Google Scholar

[14]

F. Le Roux, A topological characterization of holomorphic parabolic germs in the plane, Fund. Math., 198 (2008), 77-94.  doi: 10.4064/fm198-1-4.  Google Scholar

[15]

F. Le Roux, L'ensemble de rotation autour d'un point fixe, (French) [The rotation set around a fixed point for surface homeomorphisms], Astérisque 350 (2013), x+109pp.  Google Scholar

[16]

S. Matsumoto, Types of fixed points of index one of surface homeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 1181-1211.  doi: 10.1017/S0143385701001559.  Google Scholar

[17]

S. Matsumoto, Prime end rotation numbers of invariant seperating continua of annulus homeomorphisms, Proc Am. Math. Soc., 140 (2012), 839-845.  doi: 10.1090/S0002-9939-2011-11435-7.  Google Scholar

[18]

G. S. McCarty, Homeotopy groups, Trans. Amer. Math. Soc., 106 (1963), 293-304.  doi: 10.1090/S0002-9947-1963-0145531-9.  Google Scholar

[19]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. doi: 10.1093/oso/9780198794899.001.0001.  Google Scholar

[20]

V. A. Naǐshul', Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in $\textbf{C}^{2}$ and $\textbf{C}P^{2}$, Trudy Moskov. Mat. Obshch., 44 (1982), 235-245.   Google Scholar

[21]

J. Yan, Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms, Ergodic Theory Dynam. Systems, 36 (2016), 2293-2333.  doi: 10.1017/etds.2015.18.  Google Scholar

Figure 1.  The hyperbolic sectors
Figure 3.  A sketch map of $\mathcal{F}'$
Figure 5.  The dynamics and the foliation generated by $g(x, y) = x^2+y^2$
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