Equivariant dynamics theorem

Posted Friday March 3 2023.

Suppse you have a “spatiotemporal dynamical system”:

In addition, assume there is a group \(G\) along with a group action on \(X\) denoted by \(g \cdot x\) for \(g \in G\) and \(x \in X\). Any such action can be extended to an action on \(U\) by applying the transformation pointwise, which we can also denote by \(g \cdot u\) for \(u \in U\). Now assume the dynamical system is \(G\)-equivariant with respect to the action of \(G\), meaning it obeys the equation \[ F(t,g\cdot u)=g \cdot F(t,u) \] If \(F\) is \(G\)-equivariant, then whenever \(u\) is a constant function, so is \(F(t,u)\) for all \(t\). It follows that we can define a local action \(f:T \times Y \to Y\) by the action of \(F\) on the constant functions. Is there a name for this little \(f\)?

Edit: Here is a quote from Long-time behavior of a class of biological models by Weinberger (1982):

A constant function \(\alpha\) is clearly translation invariant. That is, \(T_y\alpha=a\) for all \(y\). Consequently, \(T_yQ[\alpha]= Q[\alpha]\) for all \(y\), which means that \(Q[\alpha]\) is again a constant. This simply states that in a homogeneous habitat the effects of migration cancel when \(u\) is constant. Thus, the properties of the model in the absence of migration can be found by looking at what \(Q\) does to constant functions.