Continuous group action

Abstract: Spiral waves are rotating waves of reaction-diffusion equations on the plane. In this Note, a center-manifold reduction for the dynamics of spiral waves is presented. Bifurcations of rigidly-rotating spiral waves are then described by ordinary differential equations, which are equivariant under the (special) Euclidean group SE(2). Several difficulties arise in this analysis because SE(2) is not compact and does not induce a strongly continuous group action on the underlying function space.

Abstract: We show that group actions on many treelike compact spaces are not too complicated dynamically.We first observe that an old argument of Seidler (1990) implies that every action of a topological group $G$on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of $G$ on a local dendron is null. We then use a more direct method to show that every continuous group action of $G$ on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees.As a related result, we show that Hellys selection principle can be extended to bounded monotone sequencesdefined on median pretrees (for example, dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.

Abstract: We prove an estimate for the difference of two solutions of the Schr\"odinger map equation for maps from $T^1$ to $S^2$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations We also prove that without taking this quotient, for any $t>0$ the flow map at time $t$ is discontinuous as a map from $\mathcal{C}^\infty(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions $(\mathcal{C}^\infty(T^1,\R^3))^*$