基于零和对策和虚拟对策的一类新的随机游动Hamilton流
A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play
摘要：
在本文中，我们与动力学相关的零和对策（虚拟游戏）的哈密顿动力学。事实证明，从虚构的发挥诱导的哈密顿动力学，具有相当不同的性质，从那些发现在更经典定义的哈密顿动力学。虽然矢量场是分段常数（因此流$ϕ_t$是分段平移），但动态是相当丰富的。例如，存在一个Hamilton $H$，使得对于每个$t>0$，水平集$H^{-1}(t)$与$S^3$同胚（水平集由$\R^4$中的超平面组成），并具有以下性质。在$H^{-1}(1)$中存在哈密尔顿流的周期轨道$Γ$，使得到在$x\in Γ$处横向于$Γ$的部分$Z\subset H^{-1}(1)$的第一返回映射$F$充当随机游动：在$Z$中存在一个嵌套的环序列$A_n$（在$x$周围，因此$\cup A_n\cup \{x\}$是$Z$中$x$的邻域）几何收缩为$x$，使得对于具有$|n(i+1)-n(i)|\le 1$的每个序列$n(i)\ge 0$，存在点$z\in Z$，使得对于所有的$i\ge 0$，存在$F^i(z)\in A_{n(i)}$。
英文摘要：
In this paper we relate dynamics associated to zero-sum games (Fictitious play) to Hamiltonian dynamics. It turns out that the Hamiltonian dynamics which is induced from fictitious play, has properties which are rather different from those found in more classically defined Hamiltonian dynamics. Although the vectorfield is piecewise constant (and so the flow $ϕ_t$ piecewise a translation), the dynamics is rather rich. For example, there exists a Hamilton $H$ so that for each $t>0$ the level set $H^{-1}(t)$ is homeomorphic to $S^3$ (the level sets consist of pieces of hyperplanes in $\R^4$) and with the following property. There exists a periodic orbit $Γ$ of the Hamiltonian flow in $H^{-1}(1)$ so that the first return map $F$ to a section $Z\subset H^{-1}(1)$ transversal to $Γ$ at $x\in Γ$ acts as a random-walk: there exist a nested sequence of annuli $A_n$ in $Z$ (around $x$ so that $\cup A_n\cup \{x\}$ is a neighbourhood of $x$ in $Z$) shrinking geometrically to $x$ so that for each sequence $n(i)\ge 0$ with $|n(i+1)-n(i)|\le 1$ there exists a point $z\in Z$ so that $F^i(z)\in A_{n(i)}$ for all $i\ge 0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
