论文标题
紧凑型符号系统的多个相变
Multiple phase transitions on compact symbolic systems
论文作者
论文摘要
令$ ϕ：x \ to \ mathbb r $是有限字母上的符号动力学系统$ t：x \ x $相关的连续潜力。引入一个参数$β> 0 $（解释为反温度），我们研究了压力函数的规律性$β\ mapsto p _ {\ rm top}（βD}（βϕ）$在$ [α，\ infty）$上，$α> 0 $。我们说，如果压力函数$ p _ {\ rm top} $在$β_0$的情况下，$β_0$的相变为$β_0$。这相当于潜在的$ β_0ϕ $具有具有不同熵的两个（Ergodic）平衡状态。对于任何$α> 0 $，以及$ [α，\ infty）$包含的任何实数$（β_n）$的任何增加的顺序，我们构建了一个潜在的$ ϕ $，其相位过渡以$ [α，\ infty）$精确地发生在$β_N$中。特别是，我们获得了具有一组无限的相变的潜力。
Let $ϕ:X\to \mathbb R$ be a continuous potential associated with a symbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing a parameter $β>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $β\mapsto P_{\rm top}(βϕ)$ on an interval $[α,\infty)$ with $α>0$. We say that $ϕ$ has a phase transition at $β_0$ if the pressure function $P_{\rm top}(βϕ)$ is not differentiable at $β_0$. This is equivalent to the condition that the potential $β_0ϕ$ has two (ergodic) equilibrium states with distinct entropies. For any $α>0$ and any increasing sequence of real numbers $(β_n)$ contained in $[α,\infty)$, we construct a potential $ϕ$ whose phase transitions in $[α,\infty)$ occur precisely at the $β_n$'s. In particular, we obtain a potential which has a countably infinite set of phase transitions.
