论文标题
与身份相切的偏斜产品的动力学
Dynamics of skew-products tangent to the identity
论文作者
论文摘要
我们研究了与身份相切的通用偏斜产品的局部动力学，即$ p（z，w）=（p（z），q（z），q（z，w））$的映射，$ dp_0 = \ mathrm {id} $。更确切地说，我们专注于起源的非分类第二差速器的地图。这样的地图具有本地正常形式$ p（z，w）=（z-z^2+o（z^3），w+w^2+bz^2+o（\ |（z，w）\ |^3））$。我们证明了抛物线域的存在，并证明在这些抛物线域内轨道在且仅当$ b \ in（\ frac {1} {4} {4} {4}，+\ infty）$时，轨道会收敛。此外，我们证明了一种抛物线内蛋白寄生虫的存在，其中重新归一化的限制与以前已知的情况不同。这有许多后果：在$ p $系数的二磷酸条件下，我们证明存在具有等级1限制地图的流浪域。我们还提供了二次偏斜产物的明确例子，这些偏斜产物具有许多宏伟的流浪域，我们给出了一个明确的例子，描述了具有fatou组分的偏斜产物图，表现出历史性的行为。最后，我们构建了各种拓扑不变的，这使我们能够回答一个问题。
We study the local dynamics of generic skew-products tangent to the identity, i.e. maps of the form $P(z,w)=(p(z), q(z,w))$ with $dP_0=\mathrm{id}$. More precisely, we focus on maps with non-degenerate second differential at the origin; such maps have local normal form $P(z,w)=(z-z^2+O(z^3),w+w^2+bz^2+O(\|(z,w)\|^3))$. We prove the existence of parabolic domains, and prove that inside these parabolic domains the orbits converge non-tangentially if and only if $b \in (\frac{1}{4},+\infty)$. Furthermore, we prove the existence of a type of parabolic implosion, in which the renormalization limits are different from previously known cases. This has a number of consequences: under a diophantine condition on coefficients of $P$, we prove the existence of wandering domains with rank 1 limit maps. We also give explicit examples of quadratic skew-products with countably many grand orbits of wandering domains, and we give an explicit example of a skew-product map with a Fatou component exhibiting historic behaviour. Finally, we construct various topological invariants, which allow us to answer a question of Abate.
