分形空间中的动力学
Dynamics in fractal spaces
摘要：
我们研究一个粒子在不可微空间中的动力学。非光滑几何对象具有固有的概率性质，因此，在其领域中的物体的运动中引入随机性。我们使用纤维丛的数学概念来描述通过不可微点的测地线轨迹的多值性质。然后，我们推广我们的概念，无处不在的非光滑结构。由此产生的理论框架可以被认为是杂交的表面理论和随机过程理论。我们尽可能地保持概念的一般性，以便引入能够模拟任何可想象的连续但不可微空间的分形或波动的其他基本过程。
英文摘要：
We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.
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