On the combinatorics of external rays in the dynamics of the complex henon map
On the combinatorics of external rays in the dynamics of the complex henon map
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复Henon图动力学中外射线的组合学
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发表时间：
1998
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通讯作者：
Ricardo A. Oliva
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作者：
Ricardo A. Oliva
我们给出了描述由复Henon映射$$f\Sb{a，c}：{\bf C}\sp2\to{\bf C}\sp2，\qquad(x，y)\mapsto(x\sp2+c-ay，x)所产生的螺线管商的组合模型。这些模型编码了Henon家族中特定映射的外部射线的标识。我们使用我们为此研究开发的计算工具，对R$\sp2$中参数空间区的结构进行了经验研究。我们给出了外部射线标识集变化引起的分叉的组合描述。我们的技术使我们能够在参数空间中检测、预测和定位分叉曲线。我们在实参数空间的一个区域中描述了一类特殊的分支族，并期望映射具有简单的动力学性质。我们计算了这个族中的前几条分支曲线，并对它们进行了组合标号。我们的计算机实验还表明，在Henon族$f\sb{a，c}$已连接Julia集$J\sb{a，c}的实参数空间区域内存在空隙。$我们证明了为什么对这个空隙的验证将意味着a的值的存在，对于该值，水平-a Mandelbrot集，$M\sb{a}=\{c\in{\bf C}：J\sb{a，c}\{\rm是连通的}，$是不连通的。
We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex Henon map$$f\sb{a,c}:{\bf C}\sp2\to{\bf C}\sp2,\qquad(x,y)\mapsto(x\sp2+c-ay,x).$$These models encode identifications of external rays for specific mappings in the Henon family. We investigate the structure of a region of parameter space in R$\sp2$ empirically, using computational tools we developed for this study. We give a combinatorial description of bifurcations arising from changes in the set of identifications of external rays. Our techniques enable us to detect, predict, and locate bifurcation curves in parameter space. We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have simple dynamics. We compute the first few bifurcation curves in this family and label them combinatorially. Our computer experiments also indicate the existence of gaps within the region of real parameter space where Henon family $f\sb{a,c}$ has connected Julia set $J\sb{a,c}.$ We show why the verification of this gap would imply the existence of values of a for which the level-a Mandelbrot set, $M\sb{a}=\{c\in{\bf C}:J\sb{a,c}\ {\rm is\ connected}\},$ is not connected.