论文标题
朱莉娅的佐里希地图
Julia sets of Zorich maps
论文作者
论文摘要
指数家族的朱莉娅集合$e_κ：z \mapstoκe^z $，$κ> 0 $在$κ> 1/e $本质上是Misiurewicz的整个复杂平面。后来，Devaney和Krych表明，朱莉娅套装是$ 0 <κ\ leq1/e $的，是成对的偏离简单曲线的无数结合，趋向于无穷大。 Bergweiler概括了Devaney和Krych的结果，用于指数图的三维类似物，称为Zorich Map。我们证明，具有对称性的某些Zorich地图的朱莉娅集是整个$ \ mathbb {r}^3 $概括Misiurewicz的结果。此外，我们表明Zorich地图的周期点在$ \ Mathbb {r}^3 $中密集，并且其逃逸集已连接，从而概括了REMPE的结果。我们还概括了Ghys，Sullivan和Goldberg定理的指数动力学。
The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb{R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.
