Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.

The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow.

Let ${\displaystyle X}$ be a topological space, and ${\displaystyle f\colon X\to X}$ a homeomorphism. If ${\displaystyle p}$ is a fixed point for ${\displaystyle f}$, the stable set of ${\displaystyle p}$ is defined by

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