Earthquake曲线切向量上的范数

本文研究了定义在earthquake曲线切向量上的范数，首先证明了一条earthquake曲线ht上初始切向量的范数等价于earthquake测度σ的Thurston范数．其次证明了当t→∞时，ht的切向量Vt的范数增长渐近等于O(||→||The^Ct||σ||Th)，其中C是正的万有常数，||σ||Th是σ的Thurston范数，而O所代表的常数是渐近万有的，也即当t||σ||Th充分大时它是万有的．此外，附带证明了定义在Zygmund有界函数上的两种交比范数是等价的．

On a Norm of Tangent Vectors to Earthquake Curves

In this paper, we study a norm defined on tangent vectors to earthquake curves.We show first that the norm of the initial tangent vector toan earthquake curve ht is equivalent to the Thurston norm of the earthquake measure σ which determines the curve. We also show that the norm of the tangent vector Vt of ht is asymptotically equal to O(‖σ‖The ct/2‖σ||Th) as t →+∞where C is a universal positive constant, ‖σ‖Th denotes the Thurston norm of σ and the constant O is asymptotically universal in the sense that it is universal if t‖σ‖Th is big enough. As a side work,we also show that two cross-ratio norms defined on Zygrnund bounded functions are also equivalent.Finally we summarizethe correspondence betweeu vanishing conditiors on σ and vanishing conditions on the initialvector V.