【作者】 吕菁；

【导师】 陈纪修； 邱维元；

【作者基本信息】 复旦大学 ， 基础数学， 2003， 博士

【摘要】 本文主要研究双二次多项式的动力系统。 在上世纪末，J．C．Yoccoz对复动力系统理论作出了重要的贡献，其中之一就是对二次多项式pc(Z)=Z~2+c的Julia集和Mandelbrot集M的局部连通性的研究。在他的工作中，Yoccoz引进了一种强有力的方法——拼图技巧。利用这个技巧，他得到了对c∈M，如果二次多项式pc(z)不是无穷可重整化的，并且没有抛物周期点，那么Julia集J(Pc)是局部连通的，并且M在c也是局部连通的。 差不多在同时，B．Branner和J.H.Hubbard研究了三次多项式的动力系统。他们用类似的拼图技巧，对三次多项式的Julia集的连通性和完全不连通性作出了完整的描述。 这种拼图技巧现在被称为Branner-Hubbard-Yoccoz拼图理论。利用这个理论，Faught和Hubbard研究了有临界不动点的单参数三次多项式族的Julia集及其连通迹的局部连通性。 上面各种情形中的多项式只有一个单临界点需要进行考虑。对于更高次多项式，譬如四次多项式，可能有多个临界点需要考虑，此时必须推广Branner-Hubbard-Yoccoz理论。A.Douadv曾建议对双二次多项式的动力系统进行研究，所谓双二次多项式就是两个二次多项式的复合，也即偶四次多项式，在一个线型共轭下可以记为 fz)=Z~4+αZ~2+b，这里o，b是参数，点0是它的一个临界点，另外还有两个对称的临界点±((-a)、2)~2。此时，有两个临界点需要考虑。但由于要考虑的两个临界点±((-a)、2)~2对称性，这使得我们有可能推广Branner-Hubbard—Yoccoz理论来处理这种情形。 我们首先通过推广Branner-Hubbatd-Yoccoz拼图理论，对双二次多项式的Julia集的连通性进行了研究，得到了与Branner和Hubbard关于三次多项式Julia集的连通性相类似的结果：定理1．设／是R一次多项式；那么（“1） f的J。l。a集是连通的当且仅当 f的所有的临界卢、的轨道都是有界的。（2 f的J。l。a集是完全不连通的当且仅当 f的任何临界声、不属于填充 Julm集的周期分支内。你 当j的临界卢、不满足条件小，”J时，f的几ha集有可数个非平凡的连通分支，这里，一个分支是平凡的，如果它仅由一个卢、组成。。 对于（3），我们进一步有：填充Juia集包含临界点的周期分支及其迭代逆象同胚于一个有连通Julia集的二次多项式的填充Julia集，而其他分支都是单点。 其次，我们讨论了临界点0为不动点的单参数双二次多项式族；该族可表示为： 人（。）＝。‘一ZC’。‘其中C为参数。在这个参数表示下，0为临界不动点；另外两个关于0点对称的临界点为士C。 我们得到的第二个结论是关于动力学平面上人的Julia集的局部连通性：定理2．设人的J，。l。a集是连通的。如果人没有抛物周期卢、；且不是可重整化的，那么人人）是局部连通的。 这里，it称为可重整化的定义为：存在单连通区域L’、U’及正整数n；使得I’〔U，仁’包含临界点c或一c，”：r一U是一个二层分支覆盖，且拟共形共轭于一个有连通Julia集的二次多项式。 进一步，我们考虑人在0点的直接吸引域，得到定理 3．＆。果人的J。l。a集是连通的，那么人在 0，卢、的直接吸引域是 Joulan曲线，从而是局部连通的。 最后，我们研究了单参数双二次多项式族（人：CEC｝的参数平面的连通迹M和捕获分支v。由定义，连通迹川是使人人）连通的参数C的集合；一个捕获分支0是使人的临界点C属于0点的直接吸引域或其迭代逆象的参数集的一个连通分支。包含0的捕获分支称为主捕获分支，记为RO。 我们的第三个结论是关于连通迹川的连通性。对于CEC＼川，设人是人关于。的B0ttcher映射，并令 WKJ”中。【V V上一上ioJ， 11其中十万一 1他称为预陪临界点。设上表示单位圆，则有定理4．申C＼川、C＼上是共形映射。因此，川是单连通的。 利用hCCOZ关于二次多项式族参数平面的拼图理论，我们研究了单参数族人在参数平面上的连通迹川和捕获分支v的局部连通性，得到定理 5．每个捕获分支的边界”都是 Jordan曲线，它们都是局部连通的。 进一步有定理6．川含有无穷多个Mandel加ot集的同胚像；并且8川在其余户、是局部连通自力。 由此可以得到；如果Al ande比rot集是局部连通的；那么o川也是局部连通的。进一步，还得到了关于连通迹川的其它一些拓扑结果。更多还原

【Abstract】 The present Ph.D dissertation is concerned with the dynamics of biquadratic polynomials.At the end of the last century, C.J.Yoccoz is made significant contributions to the theory of complex dynamics, one of which is the study of the local connectivity of the Julia sets of quadratic polynomials pc(z) = z2 + c and the Mandelbrot set M. In his work. Yoccoz introduced a powerful puzzle technique and obtained the result that if the quadratic polynomial pc(z) with , which has no indifferent cycle, is not infinitely renormalizable, then the Julia set J(pc) of pc is locally connected while dM is also locally connected at c.Almost at the same time, B.Branner and J.H.Hubbard discussed the Julia sets of cubic polynomials. With the similar technique, they gave a complete description of the connectivity and the totally disconnectivity of the Julia sets of cubic polynomials.Now the technique developed by Bianner, Hubbard and Yoccoz is called Branner-Hubbard-Yoccoz puzzle theory. A further application of the puzzle theory, due to Faught and Hubbard, is the study of the Julia sets and the connectedness locus of a cubic family with one critical fixed point.In all the above cases, the polynomials have only one critical point to be considered. For the dynamics of polynomials with higher degree, for example, the quar-tic polynomials, the Branner-Hubbard-Yoccoz puzzle theory should be extended. A.Douady had suggested to study dynamics of biquadratic polynomials. By definition, a biquadratic polynomial is the composition of two quadratic polynomials, so it is an even quartic polynomials which can be written asf(z) = z4 + az2 + 6,where a, b are parameters.It is easy to see that / has a critical point at 0 and two symmetric criticalpoints at . In this family, two critical points should be considered. However, because of the symmetry of the critical points it allows us to extend Branner-Hubbard-Yoccoz puzzle theory to study our case.The first result of our work is on the connectivity of Julia sets of biquadratic polynomials. By an extension of Branner-Hubbard-Yoccoz puzzle theory, a result analogous to Branner and Hubbard’s in the cubic case is obtained. Theorem 1. Let f be a biquadratic polynomial, then(1) the Julia set of f is connected if and only if the orbits of all critical pionts are bounded;(2) the Julia set of f is totally disconnected if and only if every connected component of the filled Julia set of f containing critical points is not periodic under iterations:(3) if f does not satisfy the conditions (1) and (2), then the Julia set of f has infinitely many non-trivial connected components, where a component is trivial if it contains only one point.In the case (3). we also have that each critical component and its inverse images under iterations are homeomorphic to the filled Julia set of a quadratic polynomial with connected Julia set. Each of the other components of A’(/) is a single point.Next, we discuss a family of biquadratic polynomials with one parameter so that the critical point 0 is a fixed point. We write this family in the formfc(z)= z4-2c2z2where c is a parameter. It has one critical fixed point 0 and two symmetric critical points 眂 under this parameterization.The second result is on the local connectivity of the Julia set of fc in its dynamical plane. We obtainTheorem 2. Let the Julia set J(fc) of fc is connected. If fc has no parabolic periodic points and is not renormalizable, then J(fc] is locally connected.Here fc is renormalizable if there are simply connected domains U and U’ and a positive integer n such that contains the critical point c or -c1 and is a quadratic-like mapping with connected Julia set.Furthermore, for the immediately attracting domain at 0, we can remove the non-renormalizable condition.Theorem 3. // the Julia set J(fc) of fc is connected, then the boundary of the immediately attracting domain of fc at 0 is a Jordan curve. Hence it is locally connected.Finally, we turn to the parameter plane of the family f更多还原