Research on the dynamics of cubic polynomials (on the topological structure of the parameter space)

研究了实三次多项式族的拉伸射线的落地性质。拉伸射线由逃逸轨迹中的拟保形变形定义，是二次多项式的Mandelbrot集合的外部射线的推广。波切尔向量是移位轨迹上每条拉伸射线上的不变量。我们证明了具有非积分波切尔向量的抛物线上的射线不着陆，即它们具有非平凡的积累集。利用抛物线内爆理论进行了证明。对于有理波切尔向量，我们也使用径向Julia集合的概念。这些带积分波切尔向量的射线以相同的法图向量落在抛物线上的点上。在抛物线弧线下面，所有的拉伸射线降落，我们已经描述了它们的落点。移动轨迹中的射线落在临界前缀的地图上。在剩余轨迹射线的证明中，我们必须使用二次多项式的双曲密度定理。

We have investigated the landing property of the stretching rays for the family of real cubic polynomials. Stretching rays are defined by a quasi-conformal deformation in the escape locus and are generalizations of external rays of the Mandelbrot set for quadratic polynomials. The Boettcher vector is an invariant on each stretching ray in the shift locus.We have proved that the rays above the parabolic arc with non-integral Boettcher vectors do not land, i.e. they have non-trivial accumulation sets. The proof is done by applying the theory of parabolic implosion. In case of rational Boettcher vectors, we also use the notion of radial Julia sets. Those rays with integral Boettcher vectors land at points on the parabolic arc with the same Fatou vectors.Below the parabolic arc, all the stretching rays land and we have characterized their landing points. The rays in the shift locus land at critically prefixed maps. In the proof for rays in the remaining locus, we have to use the theorem on density of hyperbolicity in the real quadratic polynomials.