Shizuo Nakane

ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

Formation of shocks for a single conservation law

The initial value problem for an equation of scalar conservation law in several space dimensions is considered. By the method of characteristics, the solution of this problem with

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

C^\infty

DYNAMICS OF A FAMILY OF QUADRATIC MAPS IN THE QUATERNION SPACE

-initial datum is concretely constructed. Generally, this solution becomes multivalued in finite time. By virtue of the theory of singularities of

Landing property of stretching rays for real cubic polynomials

C^\infty

Presurgical nasoalveolar molding orthopedic treatment improves the outcome of primary cheiloplasty of unilateral complete cleft lip and palate, as assessed by naris morphology and cleft gap.

-mappings, its structure as a multivalued function is completely revealed. The entropy solution is constructed by making it single-valued. In this process, shocks occur. Shock surfaces are constructed by using the stable manifold theory. Thus propagation of shocks is described.

A novel method for evaluating postsurgical results of unilateral cleft lip and palate with the use of Hausdorff distance: presurgical orthopedic treatment improves nasal symmetry after primary cheiloplasty

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials

A bifurcation phenomenon for a Dirichlet problem with an exponential nonlinearity

\bar{z}^d + c

Postcritical sets and saddle basic sets for Axiom A polynomial skew products

. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio

Branner-Hubbard-Lavaurs deformations for real cubic polynomials with a parabolic fixed point

2