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Stochastic perturbations and Ulam's method for W-shaped maps
Actions of Baumslag-Solitar groups on surfaces
1. | IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30,Montevideo |
2. | Laboratoire Paul PAINLEVÉ, Université de Lille1, 59655 Villeneuve d'Ascq Cédex, France |
This paper deals with the dynamics of actions of $BS(1, n)$ on closed orientable surfaces. We exhibit a smooth $BS(1,n)$-action without finite orbits on $\mathbb{T} ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid.
We develop a general dynamical study for faithful topological $BS(1,n)$-actions on closed surfaces $S$. We prove that such actions $ < f, h \ | \ h o f o h^{-1} = f^n >$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty.
When $S= \mathbb{T}^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of $BS(1,n)$ on $\mathbb{T}^2$.
When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ and isotopic to identity then $fix(f)$ contains any minimal set.
References:
[1] |
M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1.
doi: 10.1007/PL00004317. |
[2] |
C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61.
doi: 10.2307/1971485. |
[3] |
L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877.
doi: 10.2140/gt.2004.8.877. |
[4] |
S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075.
|
[5] |
B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001). Google Scholar |
[6] |
B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.
doi: 10.1215/S0012-7094-01-10636-4. |
[7] |
J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107.
doi: 10.2307/2001018. |
[8] |
J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441.
doi: 10.1215/S0012-7094-06-13132-0. |
[9] |
J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557.
doi: 10.1017/S0143385706001088. |
[10] |
J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443.
doi: 10.3934/jmd.2007.1.443. |
[11] |
E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.
|
[12] |
N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701.
doi: 10.2140/agt.2011.11.1701. |
[13] |
M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993).
|
[14] |
M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975). Google Scholar |
[15] |
M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977).
|
[16] |
E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97.
|
[17] |
A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395.
doi: 10.1090/S0002-9939-09-10173-9. |
[18] |
M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490.
doi: 10.1112/jlms/s2-40.3.490. |
[19] |
Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399.
|
[20] |
A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13.
doi: 10.1007/s00574-004-0002-2. |
[21] |
J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149.
doi: 10.1017/S0143385700003345. |
[22] |
J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567.
|
[23] |
J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123.
|
[24] |
M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273.
|
[25] |
M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218.
doi: 10.1007/BFb0076436. |
show all references
References:
[1] |
M. Belliart and I. Liousse, Actions sans point fixe sur les surfaces compactes,, Preprint, 34 (1994), 1.
doi: 10.1007/PL00004317. |
[2] |
C. Bonatti, Un point fixe commun pour des difféomorphismes commutants de $S^2$,, Ann. of Math., 129 (1989), 61.
doi: 10.2307/1971485. |
[3] |
L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$,, Geom. Topol., 8 (2004), 877.
doi: 10.2140/gt.2004.8.877. |
[4] |
S. Druck, F. Fang and S. Firmo, Fixed points of discrete nilpotent group actions on $S^2$,, Ann. Inst. Fourier (Grenoble), 52 (2002), 1075.
|
[5] |
B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds $I$: Actions of nonlinear groups,, Preprint (2001)., (2001). Google Scholar |
[6] |
B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups,, Duke Math. J., 106 (2001), 581.
doi: 10.1215/S0012-7094-01-10636-4. |
[7] |
J. Franks, Realizing rotation vectors for torus homeomorphisms,, Trans. Amer. Math. Soc., 311 (1989), 107.
doi: 10.2307/2001018. |
[8] |
J. Franks and M. Handel, Distortion elements in group actions on surfaces,, Duke Math. J., 131 (2006), 441.
doi: 10.1215/S0012-7094-06-13132-0. |
[9] |
J. Franks, M. Handel and K. Parwani, Fixed points of abelian actions on $S^2$,, Erg. Th. and Dyn. Sys., 27 (2007), 1557.
doi: 10.1017/S0143385706001088. |
[10] |
J. Franks, Handel and K. Parwani, Fixed points of abelian actions,, Jour. of Modern Dynamics, 1 (2007), 443.
doi: 10.3934/jmd.2007.1.443. |
[11] |
E. Ghys, Groups acting on the circle,, Enseign. Math. (2), 47 (2001), 329.
|
[12] |
N. Guelman and I. Liousse, $C^1$ actions of Baumslag Solitar groups on $S^{1}$,, Algebraic & Geometric Topology, 11 (2011), 1701.
doi: 10.2140/agt.2011.11.1701. |
[13] |
M. Gromov, Asymptotic invariants of infinite groups,, in, 2 (1993).
|
[14] |
M. Hirsch, A stable analytic foliation with only exceptional minimal set,, Lecture Notes in Math. Springer-Verlag, 468 (1975). Google Scholar |
[15] |
M. Hirsch, C. Pugh and M. Shub, Invariant manifolds,, Lecture Notes in Math. Springer-Verlag, 583 (1977).
|
[16] |
E. Lima, Common singularities of commuting vector fields on 2-manifolds,, Comment. Math. Helv., 39 (1964), 97.
|
[17] |
A. McCarthy, Rigidity of trivial actions of abelian-by-cyclic groups,, Proc. Amer. Math. Soc., 138 (2010), 1395.
doi: 10.1090/S0002-9939-09-10173-9. |
[18] |
M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori,, J. London Math. Soc., 40 (1989), 490.
doi: 10.1112/jlms/s2-40.3.490. |
[19] |
Y. Moriyama, Polycyclic groups of diffeomorphisms on the half-line,, Hokkaido Math. Jour., 23 (1994), 399.
|
[20] |
A. Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite,, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 13.
doi: 10.1007/s00574-004-0002-2. |
[21] |
J. F. Plante, Fixed points of Lie group actions on surfaces,, Ergod. Th. and Dynam. Sys., 6 (1986), 149.
doi: 10.1017/S0143385700003345. |
[22] |
J. F. Plante and W. Thurston, Polynomial growth in holonomy groups of foliations,, Comment. Math. Helv., 51 (1976), 567.
|
[23] |
J. Rebelo and R. Silva, The multiple ergodicity of nondiscrete subgroups of Dif $f^{omega}$$(S^{1})$,, Mosc. Math. J., 3 (2003), 123.
|
[24] |
M. Shub, Expanding maps,, Global Analysis Proceedings of the Symposium on Pure Mathematics, XIV (1970), 273.
|
[25] |
M. Zdun, On embedding of homeomorphisms of the circle in a continuous flow,, Iteration Theory and Its Functional Equations, 1163 (1985), 218.
doi: 10.1007/BFb0076436. |
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