CONTINUITY AND HAUSDORFF DIMENSION OF JULIA SET CONCERNING YANG-LEE ZEROS

Considering the Julia set J（Tλ） of the Yang-Lee zeros of the Potts model on the diamond hierarchical Lattice on the complex plane, the authors proved that HDJ（Tλ） 〉 1 and discussed the continuity of J（Tλ） in Hausdorff topology for λ∈R.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

On the Simple Connectivity of Fatou Components

We shall show that for certain holomorphic maps, all Fatou components are simply connected. We also discuss the relation between wandering domains and singularities for certain meromorphic maps. Received May 8, 2002, Accepted May 24, 2002     

Mating Siegel quadratic polynomials

Let be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers and . Using a new degree Blaschke product model for the dynamics of and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers and .

     

Metathesis-based synthesis of 3-methoxy α,β-unsaturated lactones: total synthesis of (R)-kavain and of the C1-C6 fragment of jerangolid D

The total synthesis of (R)-kavain and of the C1-C6 fragment of jerangolid D has been achieved in nine and seven steps, respectively, from commercially available dimethyl d-malate. A metathesis reaction of vinyl ethers and a sulfoxide-modified Julia olefination have been employed as the key steps.     

Total synthesis of (+)-ambruticin S

A convergent total synthesis of the novel antifungal agent ambruticin S (1) has been completed from the assembly of intermediates 18, 33 and 52 that served as the respective A-, B-, and C-ring precursors. The first generation approach to a potential A-ring intermediate eventuated in the synthesis of 9a via a route that featured oxidation of the dihydroxy furan 2 and elaboration of the dihydropyranone 3 derived therefrom. Although 9a served as a precursor of 31E to complete a formal synthesis of 1, there were several inefficiencies associated with the preparation of 9a. A more expedient and efficient route to an A-ring subunit was devised that commenced with the carbohydrate-derived bisacetonide aldehyde 10 and produced 18 in five steps and 46% overall yield. The synthesis of the cyclopropyl sulfone 33 was initiated with the enantioselective cyclopropanation of 19 catalyzed by Rh₂[5(S)-MEPY]₄. Ring opening of the resultant lactone 20 followed by a series of refunctionalizations gave 33 in a total of seven steps and 46% yield from 19. Coupling of the A- and B-ring precursors 18 and 33 was then achieved via a modified Julia coupling followed by deprotection and oxidation to furnish the key intermediate 35. The dihydropyran core of the C-ring subunit precursor 49 was formed from the ring closing metathesis of the diene 48, which was prepared in three steps from the known epoxide 45, followed by oxidation. A chelation-controlled addition to the methyl ketone 49 set the stage for a stereoselective [2,3]-Wittig rearrangement that delivered the alcohol 51 that was then transformed in two steps to the sulfone 52. A traditional Julia coupling of 52 and 35 proceeded with excellent stereoselectivity, and subsequent removal of the various protecting groups gave ambruticin S (1). The longest linear sequence was 13 steps and proceeded in 4.3% overall yield.     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

Julia集和等势线上的Chebyshev多项式

用P表示一个度为d的首一多项式,J_P表示它的Julia集.本文得到Julia集J_P和其等势线Γ_P(R)上的d~n-阶Chebyshev多项式,并举例说明二者并不总是相等.     

Dynamics of Polynomial Automorphisms of C~N

ＤｙｎａｍｉｃｓｏｆＰｏｌｙｎｏｍｉａｌＡｕｔｏｍｏｒｐｈｉｓｍｓｏｆＣ~Ｎ￥ＺｈａｎｇＷｅｎｊｕｎ（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，Ｐ．Ｒ．Ｃｈｉａｎ，４７５００１）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓａｓｓｉｇｎｅｄｔｏｄｉｓ?..     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.