Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.Comment: 27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicat

Harmonic maps from degenerating Riemann surfaces

We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in $W^{1,2}$ and $C^{0}$ modulo bubbles of sequences of such maps.Comment: 27 page

Development of a mobile spreadsheet-based PID control simulation system

10.1007/s10711-006-9069-9IEEE Transactions on Education492199-207IEED

Asymptotic behabior of grafting rays

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also show that the ray grafted along a weighted system of simple closed curves is at bounded distance from Teichmuller geodesic