The context may well be of assistance:
- Consider a differential equation $x'=f(x)$. Assume that $f:\mathbb R^n\to\mathbb R^n$ is continuously differentiable. Denote by $\color{red}{t\mapsto\phi_t(x)}$ the solution to the differential equation $$\frac{d}{dt}\phi_t(x)=f(\phi_t(x)),\;\phi_0(x)=x.$$ Show that $\phi_{t+s}(x)=\phi_t(\phi_s(x))$ for all $s,t\ge0$ so that $\phi_{t+s}(x)$ is defined. This is called the flow property.
The notation is indicating that we are thinking of the expression $\phi_t(x)$ as a function of $t$ (with $x$ fixed; or rather that there is a family of functions of $t$ parameterized by $x$). The notation is read "$t$ maps to $\phi_t(x)$." One may also see something such as "Define $g_x:\mathbb{R}\to\mathbb{R}^n$ by $t\mapsto \phi_t(x)$ to be the solution to the differential equation."