We give heat kernel estimates on Julia sets J（f;） for quadratic polynomials f c（z） = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a r...We give heat kernel estimates on Julia sets J（f;） for quadratic polynomials f c（z） = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.展开更多
The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z^2-2c^2)z^2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting dom...The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z^2-2c^2)z^2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.展开更多
We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m < 1. If the free critical points of f_λ are e...We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.展开更多
Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that i...Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.展开更多
牛顿-拉夫森(Newton-Raphson,NR)搜索技术是一种非线性的离散动力系统.提出了一种求解 NR 迭代函数的 Julia 点的优化模型及进化的规划求解方法.利用非线性离散动力系统在其 Julia 集出现混沌分形现象的特点,提出了一种基于 NR 搜索技...牛顿-拉夫森(Newton-Raphson,NR)搜索技术是一种非线性的离散动力系统.提出了一种求解 NR 迭代函数的 Julia 点的优化模型及进化的规划求解方法.利用非线性离散动力系统在其 Julia 集出现混沌分形现象的特点,提出了一种基于 NR 搜索技术的求解非线性方程全部根的新方法;用该方法对构成的平面四杆机构实现9个轨迹点的综合问题进行求解,表明了该方法的正确性.展开更多
文摘We give heat kernel estimates on Julia sets J（f;） for quadratic polynomials f c（z） = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10571028)
文摘The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z^2-2c^2)z^2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671091, 11731003, 11771387 and 11671092)
文摘We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.
基金Supported by National Natural Science Foundation of China(Grant Nos.10871089 and 11271179)
文摘Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.
文摘牛顿-拉夫森(Newton-Raphson,NR)搜索技术是一种非线性的离散动力系统.提出了一种求解 NR 迭代函数的 Julia 点的优化模型及进化的规划求解方法.利用非线性离散动力系统在其 Julia 集出现混沌分形现象的特点,提出了一种基于 NR 搜索技术的求解非线性方程全部根的新方法;用该方法对构成的平面四杆机构实现9个轨迹点的综合问题进行求解,表明了该方法的正确性.
