Equivariant Conley index versus degree for equivariant gradient maps (Q1942818)

This article consider the equivariant Conley index and prove that changing the equivariant Conley indices is, under some conditions, equivalent to changing the degrees for some equivariant gradient maps. Under some conditions, authors prove that the product forma \[ \nabla_G\text{-}\deg((\nabla_{\varphi_1},\nabla_{\varphi_2}),\Omega_1\times\Omega_2)=\nabla_G-\deg(\nabla_{\varphi_1},\Omega_1)\star \nabla_G-\deg(\nabla_{\varphi_2},\Omega_2) \] holds for \(\Omega_1,\Omega_2 \subset \mathbb {V}_i\) open, bounded and G-invariant subsets of a G-representation \({\mathbb{V_i}}\) and \(\varphi_i\in C_G^1(\mathbb V_i,\mathbb R)\) \(\Omega_i\) admissible functions for \(i=1,2\). Authors also present sufficient conditions for the existence of local and global bifurcations of critical orbits.