Hyperbolic Motions

Hyperbolic Automorphisms and Holomorphic Motions in C 2

Holomorphic motions have been an important tool in the study of complex dynamics in one variable. In this paper we provide one approach to using holomorphic motions in the study of complex dynamics in two variables. To introduce these ideas more fully, let 1r be the disk of radius r and center 0 in the plane, let P1 be the Riemann sphere, and recall that a holomorphic motion of a set E ⊂ P1 is ...

Limiting behaviors of the Brownian motions on hyperbolic spaces

Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. Using the representations, we show that the almost sure convergence of the Brownian motions and t...

On the Generating Motions and the Convexity of a Well-Known Curve in Hyperbolic Geometry

In Euclidean geometry the vertices P of those angles ∠APB of size α that pass through the endpoints A, B of a given segment trace the arc of a circle. In hyperbolic geometry on the other hand a set of equivalently defined points P determines a different kind of curve. In this paper the most basic property of the curve, its convexity, is established. No straight-forward proof could be found. The...

Wave Patterns and Slow Motions in Inviscid and Viscous Hyperbolic Equations with Stiff Reaction Terms

We study the behavior of solutions to the inviscid (A = 0) and the viscous (A > 0) hyperbolic conservations laws with stiff source terms ut + f(u)x = − 1 W (u) + Auxx , withW (u) being the double-well potential. The initial value problem of this equation gives, to the leading order, piecewise constants solutions connected by shock layers and rarefaction layers. In this paper we establish the la...

[Motions].