Farey映射的热力学形式
On the Thermodynamic Formalism for the Farey Map
摘要：
利用单位区间的简单映射Farey映射来描述混沌现象。通过符号动力学将映射迭代下的点的长期动力学行为转化为自旋系统。然后，可以使用来自动力系统理论和统计力学的方法来分析映射，分别是zeta函数和传递算子。间歇性被认为是有问题的分析，由于存在一个“无关紧要的固定点”。在映射迭代下的点非常缓慢地远离该点，从而产生用于计算的病态收敛时间。这个困难是通过去一个适当的诱导子系统，这也导致了一个诱导zeta函数和诱导转移算子。在那里获得的结果可以传输回原始系统。主要工作分为两个部分。第一个演示了诱导版本的zeta函数和传递算子之间的连接提供有用的结果，关于zeta函数的解析。第二部分包含了一个详细的分析的压力函数的诱导系统，因此，通过考虑的诱导zeta函数的收敛半径上的界限，原来的。特别是，渐近行为的压力函数在极限$β$，逆的“温度”，趋于负无穷大的确定和相变的存在和性质在$β=1$也进行了讨论。
英文摘要：
The chaotic phenomenon of intermittency is modeled by a simple map of the unit interval, the Farey map. The long term dynamical behaviour of a point under iteration of the map is translated into a spin system via symbolic dynamics. Methods from dynamical systems theory and statistical mechanics may then be used to analyse the map, respectively the zeta function and the transfer operator. Intermittency is seen to be problematic to analyze due to the presence of an `indifferent fixed point'. Points under iteration of the map move away from this point extremely slowly creating pathological convergence times for calculations. This difficulty is removed by going to an appropriate induced subsystem, which also leads to an induced zeta function and an induced transfer operator. Results obtained there can be transferred back to the original system. The main work is then divided into two sections. The first demonstrates a connection between the induced versions of the zeta function and the transfer operator providing useful results regarding the analyticity of the zeta function. The second section contains a detailed analysis of the pressure function for the induced system and hence the original by considering bounds on the radius of convergence of the induced zeta function. In particular, the asymptotic behaviour of the pressure function in the limit $β$, the inverse of `temperature', tends to negative infinity is determined and the existence and nature of a phase transition at $β=1$ is also discussed.
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