One-parameter group of self-maps

Definition

Let $X$ be a set. A one-parameter group on $X$, or a one-parameter group of self-maps on $X$, is a homomorphism from $\mathbb{R}$ to the group of all self-maps from $X$ to $X$ (i.e. the symmetric group on $X$ as a set).

Particular cases of interest are:

• One-parameter group of self-homeomorphisms
• One-parameter group of self-diffeomorphisms