Author:
Chen Qinbo,Damjanović Danijela,Petković Boris
Abstract
AbstractLet $$\mathcal {F}$$
F
and $$\mathcal {K}$$
K
be commuting $$C^\infty $$
C
∞
diffeomorphisms of the cylinder $$\mathbb {T}\times \mathbb {R}$$
T
×
R
that are, respectively, close to $$\mathcal {F}_0 (x, y)=(x+\omega (y), y)$$
F
0
(
x
,
y
)
=
(
x
+
ω
(
y
)
,
y
)
and $$T_\alpha (x, y)=(x+\alpha , y)$$
T
α
(
x
,
y
)
=
(
x
+
α
,
y
)
, where $$\omega (y)$$
ω
(
y
)
is non-degenerate and $$\alpha $$
α
is Diophantine. Using the KAM iterative scheme for the group action we show that $$\mathcal {F}$$
F
and $$\mathcal {K}$$
K
are simultaneously $$C^\infty $$
C
∞
-linearizable if $$\mathcal {F}$$
F
has the intersection property (including the exact symplectic maps) and $$\mathcal {K}$$
K
satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $$\mathbb {Z}^2$$
Z
2
-actions on the cylinder, generated by commuting twist maps.
Funder
Royal Institute of Technology
Publisher
Springer Science and Business Media LLC
Reference38 articles.
1. Avila, A., Fayad, B., Le Calvez, P., Disheng, X., Zhang, Z.: On mixing diffeomorphisms of the disc. Invent. Math. 220(3), 673–714 (2020)
2. Anosov, D.V., Katok, A.B.: New examples of ergodic diffeomorphisms of smooth manifolds. Uspehi Mat. Nauk 25(4(154)), 173–174 (1970)
3. Sylvain Crovisier. Exotic rotations. Course Notes, 2006
4. Chong-Qing C., Yi-Sui S.: Existence of invariant tori in three-dimensional measure-preserving mappings. Celestial Mech. Dyn. Astronom. 47(3), 275–292 (1989/90)
5. Damjanović, D., Fayad, B.: On local rigidity of partially hyperbolic affine $$\mathbb{Z}^k$$ actions. J. Reine Angew. Math. 751, 1–26 (2019)
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献