On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov’s Theorem

In this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the $${\mathcal {C}}^\infty $$ C ∞ case, where we show that only versality is possible.

Analytic moduli for unfoldings of germs of generic analytic diffeomorphisms with a codimension parabolic point

In this paper we provide a complete modulus of analytic classification for germs of generic analytic families of diffeomorphisms which unfold a parabolic fixed point of codimension$k$. We start by showing that a generic family can be ‘prepared’, i.e. brought to a prenormal form${f}_{\epsilon } (z)$in which the multi-parameter$\epsilon $is almost canonical (up to an action of$ \mathbb{Z} / k \mathbb{Z} $). As in the codimension one case treated in P. Mardešić, R. Roussarie and C. Rousseau [Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms.Mosc. Math. J. 4(2004), 455–498], we show that the Ecalle–Voronin modulus can be unfolded to give a complete modulus for such germs. For this purpose, we define unfolded sectors in$z$-space that constitute natural domains on which the map${f}_{\epsilon } $can be brought to normal form in an almost unique way. The comparison of these normalizing changes of coordinates on the different sectors forms the analytic part of the modulus. This construction is performed on sectors in the multi-parameter space$\epsilon $such that the closure of their union provides a neighborhood of the origin in parameter space.

Cubic polynomials with a parabolic point

We consider cubic polynomials with a simple parabolic fixed point of multiplier 1. For those maps, we prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve (except for the polynomial z+z3 where it consists in two Jordan curves). Moreover, we give a description of the dynamics and obtain the local connectivity of the Julia set under some assumptions.

Analytical determination of parabolic points on slowness surface and swallowtail points on wave surface of cubic crystals

Slowness surface for bulk wave propagation in anisotropic media can be divided into concave, saddle and convex regions by the parabolic lines. When a parabolic line crosses a symmetry plane, it leaves either an inflection point or a parabolic point. Surface normal at these points is associated with cuspidal point and swallowtail point, respectively, on the wave surface and in phonon focusing patterns. By examining the degeneracies in the Stroh eigenvalue equation, we have calculated the cuspidal points in cubic crystals analytically. In this work, the parabolic point and its surface normal are discussed. The main idea is to establish a connection between the parabolic point and the extraordinary transonic state that is related to a degeneracy with a multiplicity of four in the Stroh eigenvalue equation. Such a connection yields a series of simple expressions, which determine the locations of parabolic points and the corresponding swallowtail points. The result is demonstrated using phonon focusing patterns of cubic crystals, and the method also provides a tool for general discussion of the slowness surface geometry.