Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ...Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.展开更多
The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps ...The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps into convex balls.展开更多
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 obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥2 into any complex space Z. After lifting to the normalization of the subvariety F (Ω) Z, we...We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥2 into any complex space Z. After lifting to the normalization of the subvariety F (Ω) Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou’s theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f : Ω→ D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described.展开更多
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
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.
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
Let/(z) be a holomorph.self-map on C.-G-(0) with essential singularities 0 and It is proved that f(z) has a completdy invariant domain.D.F(f),then D is doubly connected and D contains all the singularities of the inv...Let/(z) be a holomorph.self-map on C.-G-(0) with essential singularities 0 and It is proved that f(z) has a completdy invariant domain.D.F(f),then D is doubly connected and D contains all the singularities of the inverse of f(z),moreover,if f is of the finite type, then D=F(f). This result implies that f(z) has at most one completely invariant domain in F(f).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
This article studies the inverse image of rational functions. Several theorems are obtained on the Julia set expressed by the inverse image, and a mistake is pointed out in H.Brolin＇ theorem incidentally.
Let g and h be two transcendental entire functions. Suppose that the Fatou set F（goh） contains multiply connected components. In this article, we will consider the growth of the functions g and h.
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
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
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 f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is...Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C . 展开更多
The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of ...The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of these sets are described completely.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11261002 and 11261069)Natural Science Foundation of Yunnan Province of China(Grant No.2012FZ167)Educational Commission of Yunnan Province of China(Grant No.2012Z121)
文摘Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.
文摘The author considers harmonic maps on complete noncompact manifolds, solves the Dirichlet problem in manifolds with nonnegative sectional curvature out of a compact set, and proves the Fatou theorem for harmonic maps into convex balls.
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supported by the GRF7032/08P of the HKRGC, Hong KongNational Natural Science Foundation of China (Grant No. 10971156)
文摘We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥2 into any complex space Z. After lifting to the normalization of the subvariety F (Ω) Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou’s theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f : Ω→ D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described.
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.
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.
文摘Let/(z) be a holomorph.self-map on C.-G-(0) with essential singularities 0 and It is proved that f(z) has a completdy invariant domain.D.F(f),then D is doubly connected and D contains all the singularities of the inverse of f(z),moreover,if f is of the finite type, then D=F(f). This result implies that f(z) has at most one completely invariant domain in F(f).
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.
基金Project Supported by the National Natural Science Foundation of China(10471048)the Research Fund for the Doctoral Program of Higher Education(20050574002)
文摘This article studies the inverse image of rational functions. Several theorems are obtained on the Julia set expressed by the inverse image, and a mistake is pointed out in H.Brolin＇ theorem incidentally.
文摘Let g and h be two transcendental entire functions. Suppose that the Fatou set F（goh） contains multiply connected components. In this article, we will consider the growth of the functions g and h.
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.
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).
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 f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .
基金This work was supported by the 973 Project Foundation of China.
文摘The sets of the points corresponding to the complexphases of the Potts model on the diamond hierarchcal lattice arestudied. These sets are the Fatou sets of a family of rationalmappings. The topological structures of these sets are described completely.
