An S1-degree and S1-maps between representation spheres

Applied equivariant degree, part I: An axiomatic approach to primary degree

“…It should be pointed out that the primary degree was introduced in [13] independently of [17], using the so-called regular normal approximations and winding numbers of their restrictions to normal slices around the orbits of zeros (cf. [10,11,23], where the case G = S 1 was considered). However, it is well-known (see, for instance, [21,36]) that the winding number admits an axiomatic definition as an integer-valued function satisfying homotopy, additivity and normalization properties.…”

Guaranteed estimates for the length of branches of periodic orbits for equivariant Hopf bifurcation

“…To verify condition (P2)(iii), one has to compute multiplicities of the roots of Λ H ϕ 1 (α ± , τ, β) (cf. (7)). To this end, decompose 1 V into its Γ × S 1 -isotypical components…”

“…This contradicts the usual "absence of the steady state bifurcation" condition. However, we overcome this by considering (7) instead of (6) with properly chosen H ϕ and applying Theorem 3.3. In the following example complex eigenvalues cross the imaginary axis transversally at the bifurcation point α = α 0 .…”

“…The paper is organized as follows. Section 2 contains a brief account of the S 1 -degree [7], which is the main equivariant topological tool used in the proofs (see also [4] for the axiomatic approach to the S 1 -degree and [4,15] for a systematic exposition of the equivariant degree theory and its applications to symmetric Hopf bifurcation). The main result and its proof are presented in Sections 3 and 4.…”

A short treatise on the equivariant degree theory and its applications

“…The S 1 -degree was constructed in [36,37] independently of the works [58,59] (where a more general setting was considered). For a detailed exposition of this concept based on the use of regular normal approximations we refer the reader to [68] (cf.…”