Equivariant bifurcations in four-dimensional fixed point spaces

Abstract

In honor of Marty Golubitsky on the occasion of his seventieth birthday. AbstractIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach [14] and Lauterbach and Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.

Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of S k

“…This non-gradient behaviour suggests there may well exist absolutely irreducible representations (V, G) for which Property (G) fails and the eigenvalues of DQ u along Ru are all either zero or pure imaginary. A natural place to look is the works of Lauterbach and Matthews [34], Lauterbach [33], and Lauterbach and Schwenker [35] on low dimensional families of absolutely irreducible representations with no odd dimensional fixed point spaces. However, these families do not have quadratic equivariants and the question appears open.…”

Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

“…In this case, v = 0 v 1 · · · v 10 , v 1 + · · · + v 10 = 0 (22) However, this lemma does not exclude the possibility of other solution types, and little is known in general about fixed point subspaces of even dimensions: such solutions have been found in some systems (for example, [28]), but there is currently not a general theory guaranteeing or ruling out such solutions [29]. In this system, at least one branch corresponds to a subgroup Σ for which dim Fix(Σ) = 2: the 2-3-5 branch.…”