坎农-瑟斯顿映射的奇异性
Singularity of Cannon-Thurston maps
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中文总结 AI 辅助
研究封闭纤维化双曲3-流形中坎农-瑟斯顿映射，利用测地线性质证明圆上多种自然测度经此映射到球面上会变奇异，通过对比双曲测地线靠近纤维时间比例得出结论，还对特定圆测度证明了有效结果。
中文摘要 AI 辅助
在一个封闭的纤维化双曲3-流形M中，纤维S的包含（S和M提升到万有覆盖）给出了双曲平面到双曲3-空间的指数扭曲嵌入。坎农和瑟斯顿证明存在从双曲平面的无穷远圆到双曲3-空间的无穷远2-球面的映射，该映射是满射、有限对一且给出空间填充曲线。本文利用测地线性质证明，当通过坎农-瑟斯顿映射向前推时，圆上许多自然测度相对于球面上许多自然测度变得奇异。考虑的圆测度有勒贝格测度和源于曲面群上全支撑随机游走的平稳测度，球面上的测度有勒贝格测度和源于3-流形群上几何随机游走的平稳测度。通过典型测地线的性质得到测度的奇异性，证明了相对于前推测度采样的双曲测地线渐近地有一定比例的时间靠近纤维，而相对于球面上自然测度采样的双曲测地线靠近纤维的时间比例渐近可忽略。对于更受限的一类圆测度，即勒贝格测度和源于曲面群上几何随机游走的平稳测度，还证明了靠近纤维时间比例的有效结果。
英文摘要
In a closed fibered hyperbolic 3-manifold M ,the inclusion of a fiber S, with S and M lifted to the universal covers, gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we use properties of geodesics to prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. The measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3- manifold group. We obtain the singularity of measures from the following properties of typical geodesics. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of circle measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber.
