AI 中文总结
研究离散和连续时间动力系统中触发相变的机制，证明简单拓扑条件，表明相变在多种情况典型，如\(C^1\)系统非传递时、低维非阿诺索夫系统及特定\(C^0\)流形上同胚中普遍存在相变。
AI 中文摘要
在这项工作中，我们研究了离散时间和连续时间动力系统中触发相变的机制。我们证明了一个简单的拓扑条件，它意味着存在显示相变的Hölder连续势。结果表明，相变在几种情况下是典型的。具体而言，对于\(C^1\)微分同胚和\(C^1\)向量场，非传递系统中相变是普遍存在的。在低维情况下，非阿诺索夫系统中相变是普遍的。最后，在\(C^0\)设置下，我们证明在任何不同于4维的紧致拓扑流形上，存在一个具有有限熵且显示相变的同胚密集集。
英文摘要
In this work, we investigate the mechanisms that trigger phase transitions in both discrete-time and continuous-time dynamical systems. We prove a simple topological condition that implies the existence of Hölder continuous potentials displaying phase transitions. As a consequence, we show that phase transitions are typical in several scenarios. Specifically, for $C^1$ diffeomorphisms and $C^1$ vector fields, we show that phase transitions are generic among non-transitive systems. In low dimensions, we conclude that they are generic for non-Anosov systems. Finally, in the $C^0$ setting, we prove that on any compact topological manifold of dimension different from 4, there exists a dense set of homeomorphisms with finite entropy that display phase transitions.
