有限域上可逆有理映射的组合模型
A combinatorial model for reversible rational maps over finite fields
摘要：
我们研究有限相空间动力系统的时间反演对称性，并将其应用于有限域上的双有理映射。对于具有单个反向对称族的多项式自同构，通用（即，本文证明了在大场极限下约化映象的归一化圈长存在一个分布函数R（x）=1-e^{-x}（1+x）。G. Roberts和F. Vivaldi，Nonlinearity 18（2005）2171-2192）。我们表明，这些统计量对应于两个随机对合的组成，有适当数量的不动点。这个模型也解释了实验观察，渐近地，几乎所有的周期是对称的，并且重复周期的发生概率由泊松定律决定。
英文摘要：
We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18 (2005) 2171-2192). We show that these statistics correspond to those of a composition of two random involutions, having an appropriate number of fixed points. This model also explains the experimental observation that, asymptotically, almost all cycles are symmetrical, and that the probability of occurrence of repeated periods is governed by a Poisson law.
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