Symbolic dynamics for geodesic floes
Symbolic dynamics for geodesic floes
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测地浮冰的符号动力学
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发表时间：
1981
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通讯作者：
B. Buszewski
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作者：
B. Buszewski
符号动力学在现代动力系统理论中具有中心重要性。使用符号序列来研究测地线的动力学特性起源于Koebe[21,22]和Morse[24,25]的工作，并在Hadamard[15]和Jordan[15]中已经有了预示。Koebe和Morse的方法是对负曲率曲面M上的测地线进行编码，通过记录它在M上穿过给定标记曲线集的顺序。Morse的处理允许可变曲率，但假设至少有两个边界分量，而Koebe假设恒定曲率，但允许无限连通性和不可定向性，并处理更困难的封闭曲面情况。最后一种情况是通过记录固定裤子分解的交叉点来处理的，预测简单曲线的Thurston参数化，如[11]中所述。Koebe和Morse都用他们的编码证明了存在可数多个封闭测地线和无处不在的密集（传递）测地线。莫尔斯进一步构造了第一个循环的非周期不连续运动的非合成例子（在现代术语中，一个极小无处密集集）。后来[26]Morse用规则的4</-gons处理了与盘D镶嵌相关的g属的特殊闭面。在这样的镶嵌中，区域的每条边都可以用等距来标记，等距将其粘合到同一区域的另一边，形成商sin面M。作为标记出现的等距集生成TTI(M)。因此，任何测地线都被编码为iri(M)的生成器的双重无限序列。同样的方法也适用于与任何紫红色组相关的镶嵌。我们称这样得到的序列为切割序列，并将这类生成集称为几何生成集。当然，困难在于精确地确定出现的序列的类别。对于有边界的曲面，可以得到精确约简的-序列^in^the=generators^ in^ generaLthe=problem4s=GompliGated1=henGe=在[26]中遇到的^困难（参见下面的定理3）。对于模曲面H/SL(2,2)，适当的展开式是连分式，对于上面的对称g属曲面，适当的展开式是[27]的Nielsen边界展开式。
The subject of symbolic dynamics is of central importance in the modern theory of dynamical systems. The use of symbolic sequences to study dynamical properties of geodesies originates in the work of Koebe [21, 22] and Morse [24, 25] and is already foreshadowed in Hadamard [15] and Jordan [19]. The method of Koebe and of Morse is to code a geodesic on a surface M of negative curvature by recording the order in which it traverses a given set of labelled curves on M. The treatment of Morse allows variable curvature but assumes at least two boundary components, whereas Koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface. This last case is handled by recording crossings of a fixed pants decomposition, anticipating the Thurston parameterization of simple curves as described in [11]. Both Koebe and Morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense (transitive) geodesies. Morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion (in modern terminology, a minimal nowhere dense set). Later [26] Morse treated the special closed surfaces of genus g associated to tesselations of the disc D by regular 4</-gons. Each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin-face M. The set of isometries appearing as labels generate TTI(M). Thus any geodesic is coded as a doubly infinite sequence of generators of iri(M). The same method applies quite generally to tesselations associated to any Fuchsian group. We call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric. The difficulty of course is to determine precisely the class of sequences which occur. For a surface with boundary, one obtains exactly reduced -sequences^in^the=generators^In^generaLthe=problem4s=GompliGated1=henGe=the^ difficulties encountered in [26] (see also Theorem 3 below). There is another method of coding geodesies, using certain boundary expansions for points at infinity in the universal cover of M. For the modular surface H/SL(2,2) the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the Nielsen boundary expansions of [27].