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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 |
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.
Keywords:
Surface homeomorphism,
transverse foliation,
maximal isotopy,
rotation number,
local rotation set.
Mathematics Subject Classification:
Primary: 37E30, 37E45.
Citation:
Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete and Continuous Dynamical Systems,
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. |
| [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. |
| [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. |
| [4] |
J. M. Gambaudo, P. 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.
|
| [5] |
M. E. Hamstrom,
Homotopy groups of the space of homeomorphisms on a 2-manifold, Illinois J. Math., 10 (1966), 563-573.
|
| [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. |
| [7] |
A. Koropecki, P. Le Calvez and M. Nassiri,
Prime ends rotation numbers and periodic points, Duke Math. J., 164 (2015), 403-472.
doi: 10.1215/00127094-2861386. |
| [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. |
| [9] |
A. Koropecki and F. A. Tal,
Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.
doi: 10.1007/s00222-013-0470-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
G. S. McCarty,
Homeotopy groups, Trans. Amer. Math. Soc., 106 (1963), 293-304.
doi: 10.1090/S0002-9947-1963-0145531-9. |
| [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. |
| [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.
|
| [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. |
show all references
References:
| [1] |
F. Béguin, S. Crovisier and F. Le Roux, Fixed point sets of isotopies on surfaces, preprint, arXiv: 1610.00686v2. |
| [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. |
| [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. |
| [4] |
J. M. Gambaudo, P. 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.
|
| [5] |
M. E. Hamstrom,
Homotopy groups of the space of homeomorphisms on a 2-manifold, Illinois J. Math., 10 (1966), 563-573.
|
| [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. |
| [7] |
A. Koropecki, P. Le Calvez and M. Nassiri,
Prime ends rotation numbers and periodic points, Duke Math. J., 164 (2015), 403-472.
doi: 10.1215/00127094-2861386. |
| [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. |
| [9] |
A. Koropecki and F. A. Tal,
Strictly toral dynamics, Invent. Math., 196 (2014), 339-381.
doi: 10.1007/s00222-013-0470-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
G. S. McCarty,
Homeotopy groups, Trans. Amer. Math. Soc., 106 (1963), 293-304.
doi: 10.1090/S0002-9947-1963-0145531-9. |
| [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. |
| [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.
|
| [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. |
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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