In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia se...In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.展开更多
We shall show that for certain holomorphic maps, all Fatou components are simply con- nected. We also discuss the relation between wandering domains and singularities for certain mero- morphic maps.
Suppose that P=(p\-1, p\-2, ..., p\-M)\% is a probability vector with p\-i>0 and Y={1, 2, ..., M}. Let (Y, 2\+Y, μ) be a probability space with μ(i)=p\-i, i=1, 2, ..., M, and (∑\-M, B, m)= Π \+∞\-0(Y, 2\+U, μ...Suppose that P=(p\-1, p\-2, ..., p\-M)\% is a probability vector with p\-i>0 and Y={1, 2, ..., M}. Let (Y, 2\+Y, μ) be a probability space with μ(i)=p\-i, i=1, 2, ..., M, and (∑\-M, B, m)= Π \+∞\-0(Y, 2\+U, μ). It is shown that for any a \%(0≤a ≤1) \%, there exists a set U∈B such that m(U)=a and the Julia set associated with U is equal to the Julia set associated with ∑\-M\%. Moreover repelling fixed points with respect to U are dense in the Julia set associated with U.展开更多
For a sequence (cn) of complex numbers, the quadratic polynomials fcn:= z2 + Cn and the sequence (Fn) of iterates Fn: = fcn ο ? ο fc1 are considered. The Fatou set F(Cn) is defined as the set of all $z \in \hat {\ma...For a sequence (cn) of complex numbers, the quadratic polynomials fcn:= z2 + Cn and the sequence (Fn) of iterates Fn: = fcn ο ? ο fc1 are considered. The Fatou set F(Cn) is defined as the set of all $z \in \hat {\mathbb{C}}: = {\mathbb{C}} \cup \left\{ \infty \right\}$ such that (Fn) is normal in some neighbourhood of z, while the complement J(Cn) of F(cn) (in $\hat {\mathbb{C}}$ ) is called the Julia set. The aim of this paper is to study the stability of the Julia set J(Cn) in the case where (cn) is bounded. A problem put forward by Brück is solved.展开更多
We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1（Cp） to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich...We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1（Cp） to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in Cp have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.展开更多
We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of thes...We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of these points and components is called the residual Julia set, denoted by , and is defined to be the subset of those points of the Julia set, chaotic set, which do not belong to the boundary of any component of the Fatou set (stable set). The points of are called buried points and the components of are called buried components. In this paper we extend some results related with the residual Julia set of transcendental meromorphic functions to functions which are meromorphic outside a compact countable set of essential singularities. We give some conditions where .展开更多
Let f and g be two permutable transcendental holomorphic maps in the plane.We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is o...Let f and g be two permutable transcendental holomorphic maps in the plane.We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.展开更多
Let fj M (j = 1, 2, …, m; m1) and %f be the skew product associated with the generator system {f1, f2, …, fm}. Then F(%f) is completely invariant under (%f); J(%f) is completely invariant under%f; J(%f) is perfect;...Let fj M (j = 1, 2, …, m; m1) and %f be the skew product associated with the generator system {f1, f2, …, fm}. Then F(%f) is completely invariant under (%f); J(%f) is completely invariant under%f; J(%f) is perfect; J(%f) has interior points if and only if F(%f) =; if fj MAp (p5), j = 1, 2, …, m, then the set of the repelling fixed points of%fof all orders are dense in J(%f).展开更多
In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immed...In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.展开更多
基金Supported by National Natural Science Foundation of China (Grant Nos. 10831004, 10871047)Science and Technology Commission of Shanghai Municipality (NSF Grant 10ZR1403700)
文摘In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.
基金The authors are supported by NSFC the 973 Project
文摘We shall show that for certain holomorphic maps, all Fatou components are simply con- nected. We also discuss the relation between wandering domains and singularities for certain mero- morphic maps.
文摘Suppose that P=(p\-1, p\-2, ..., p\-M)\% is a probability vector with p\-i>0 and Y={1, 2, ..., M}. Let (Y, 2\+Y, μ) be a probability space with μ(i)=p\-i, i=1, 2, ..., M, and (∑\-M, B, m)= Π \+∞\-0(Y, 2\+U, μ). It is shown that for any a \%(0≤a ≤1) \%, there exists a set U∈B such that m(U)=a and the Julia set associated with U is equal to the Julia set associated with ∑\-M\%. Moreover repelling fixed points with respect to U are dense in the Julia set associated with U.
文摘For a sequence (cn) of complex numbers, the quadratic polynomials fcn:= z2 + Cn and the sequence (Fn) of iterates Fn: = fcn ο ? ο fc1 are considered. The Fatou set F(Cn) is defined as the set of all $z \in \hat {\mathbb{C}}: = {\mathbb{C}} \cup \left\{ \infty \right\}$ such that (Fn) is normal in some neighbourhood of z, while the complement J(Cn) of F(cn) (in $\hat {\mathbb{C}}$ ) is called the Julia set. The aim of this paper is to study the stability of the Julia set J(Cn) in the case where (cn) is bounded. A problem put forward by Brück is solved.
基金Supported by National Natural Science Foundation of China (Grant Nos.10831008 and 11231009)
文摘We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1（Cp） to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in Cp have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.
文摘We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of these points and components is called the residual Julia set, denoted by , and is defined to be the subset of those points of the Julia set, chaotic set, which do not belong to the boundary of any component of the Fatou set (stable set). The points of are called buried points and the components of are called buried components. In this paper we extend some results related with the residual Julia set of transcendental meromorphic functions to functions which are meromorphic outside a compact countable set of essential singularities. We give some conditions where .
文摘Let f and g be two permutable transcendental holomorphic maps in the plane.We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.
文摘Let fj M (j = 1, 2, …, m; m1) and %f be the skew product associated with the generator system {f1, f2, …, fm}. Then F(%f) is completely invariant under (%f); J(%f) is completely invariant under%f; J(%f) is perfect; J(%f) has interior points if and only if F(%f) =; if fj MAp (p5), j = 1, 2, …, m, then the set of the repelling fixed points of%fof all orders are dense in J(%f).
基金Supported by the Scientific Research Fund of Hunan Provincial Education Department (Grant No06C245)
文摘In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.
