Polynomial diffeomorphisms ofC2. IV: The measure of maximal entropy and laminar currents
Polynomial diffeomorphisms ofC2. IV: The measure of maximal entropy and laminar currents
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C2 的多项式微分同胚。
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发表时间：
1992
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通讯作者：
J. Smillie
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作者：
E. Bedford;M. Lyubich;J. Smillie
显示有趣行为的最简单的全纯动力系统是C.这些地图的动力学研究始于20世纪20年代的Fatou和Julia，目前是一个非常活跃的研究领域。如果我们有兴趣研究可逆全纯动力系统，那么具有有趣行为的最简单的例子可能是C的多项式复同态。这些映射f：C → C使得f和f的坐标函数是全纯多项式。对于C的多项式映射，多项式的代数次数是一个有用的动力学不变量。特别地，唯一动态有趣的映射是那些度d大于1的映射。对于多项式代数同态，我们可以将代数次数定义为坐标函数的次数的最大值。然而，这不是共轭不变量。弗里德兰和米尔诺[FM]给出了另一种定义的正整数度F这是更自然的从动态的角度来看。若deg f > 1，则deg f与f的共轭类中的一个单同态的最小代数次数一致与C的多项式映射一样，deg（f）= 1的多项式同态f是相当无趣的。我们假设deg（f）> 1。对于C的多项式映射，无穷远点是吸引子。因此，“经常性”的动力学只能发生在由有界轨道组成的集合K上。一个正常的家庭的论点表明，没有扩展的内部K，所以“混沌”动力学只能发生在J = K。这个集合被称为Julia集，在多项式映射的研究中起着重要作用。对于C的同构，对象K和J中的每一个都有三个类似物。对应于一维中的集合K，我们有集合K（resp. K）由其轨道在前向有界的点组成（分别为）。后向）时间和集合K：= K <$K由具有有界全轨道的点组成。这些集合中的每一个都是不变的，K是紧的。与一维情形一样，递归只能发生在集合K上。对应于一维的集合J，我们有集合J：= K，和集合J：= J J。这些集合中的每一个都是不变的，J是紧的。一个正规族论证表明，K的内部不存在“向前”不稳定性，也不存在“向后”不稳定性。因此，“混沌”动力学，即在向前和向后时间都不稳定的循环动力学，只能发生在集合J上。Fatou和Julia在一维中使用的技术是基于Montel的正规族理论，并且不容易推广到更高的维度。一个不同的工具出现在工作布罗林[Br]，谁利用了理论的对数
The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C. The dynamical study of these maps began with Fatou and Julia in the 1920’s and is currently a very active area of research. If we are interested in studying invertible holomorphic dynamical systems, then the simplest examples with interesting behavior are probably the polynomial diffeomorphisms ofC. These are maps f : C → C such that the coordinate functions of f and f are holomorphic polynomials. For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant. In particular the only dynamically interesting maps are those with degree d greater than one. For polynomial diffeomorphisms we can define the algebraic degree to be the maximum of the degrees of the coordinate functions. This is not, however, a conjugacy invariant. Friedland and Milnor [FM] gave an alternative definition of a positive integer deg f which is more natural from a dynamical point of view. If deg f > 1, then deg f coincides with the minimal algebraic degree of a diffeomorphism in the conjugacy class of f . As in the case of polynomial maps of C, the polynomial diffeomorphisms f with deg(f) = 1 are rather uninteresting. We will make the standing assumption that deg(f) > 1. For a polynomial map of C the point at infinity is an attractor. Thus the “recurrent” dynamics can take place only on the set K consisting of bounded orbits. A normal families argument shows that there is no expansion on the interior of K so “chaotic” dynamics can occur only on J = ∂K. This set is called the Julia set and plays a major role in the study of polynomial maps. For diffeomorphisms ofC each of the objectsK and J has three analogs. Corresponding to the set K in one dimension, we have the sets K (resp. K) consisting of the points whose orbits are bounded in forward (resp. backward) time and the set K := K ∩ K consisting of points with bounded total orbits. Each of these sets is invariant and K is compact. As is in the one dimensional case, recurrence can occur only on the set K. Corresponding to the set J in dimension one, we have the sets J := ∂K, and the set J := J ∩ J. Each of these sets is invariant and J is compact. A normal families argument shows that there is no “forward” instability in the interior of K and no “backward” instability in the interior of K. Thus “chaotic” dynamics, that is recurrent dynamics with instability in both forward and backward time, can occur only on the set J . The techniques that Fatou and Julia used in one dimension are based on Montel’s theory of normal families and do not readily generalize to higher dimensions. A different tool appears in the work of Brolin [Br], who made use of the theory of the logarithmic