The Geometric Model and Coarse Lipschitz Equivalence Direct from Teichmüller Geodesics

The Ending Laminations Theorem direct from Teichmüller Geodesics

A proof of the Ending Laminations Theorem is given which uses Teichmüller geodesics directly.

Large Scale Rank of Teichmüller Space

Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmüller space equipped with either the Teichmüller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R is locally near a standard model of a flat in X . As a consequence, we sh...

On Reflections in Jordan Curves

A purely geometric method for constructing reflections in Jordan curves on the Riemann sphere based on hyperbolic geodesics is introduced. It is then possible to investigate the relations between the geometry of a Jordan domain D and the properties of the reflection by studying properties of hyperbolic geodesics. This idea is used to characterize unbounded Jordan John domains in terms of reflec...

Teichmüller geodesics of infinite complexity

Geometric Modeling of Dubins Airplane Movement and its Metric

The time-optimal trajectory for an airplane from some starting point to some final point is studied by many authors. Here, we consider the extension of robot planer motion of Dubins model in three dimensional spaces. In this model, the system has independent bounded control over both the altitude velocity and the turning rate of airplane movement in a non-obstacle space. Here, in this paper a g...