scholarly journals Analytic classification of germs of parabolic antiholomorphic diffeomorphisms of codimension k

2021 ◽  
pp. 1-37
Author(s):  
JONATHAN GODIN ◽  
CHRISTIANE ROUSSEAU
Keyword(s):  
Fixed Point ◽  
Modulus Space ◽  
Normal Form ◽  
Complete Classification ◽  
Invariant Curves ◽  
Analytic Classification ◽  
Real Analytic ◽  
Analytic Invariant ◽  
Parabolic Fixed Point

Abstract We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

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

2013 ◽  
Vol 35 (1) ◽  
pp. 274-292 ◽  
Author(s):  
C. ROUSSEAU
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Parameter Space ◽  
Parabolic Point ◽  
Codimension One ◽  
Analytic Classification ◽  
Natural Domains ◽  
Generic Family ◽  
Parabolic Fixed Point

AbstractIn 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.

scholarly journals Invariant curves around a parabolic fixed point at infinity

1990 ◽  
Vol 10 (2) ◽  
pp. 209-229 ◽  
Author(s):  
Dov Aharonov ◽  
Uri Elias
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Invariant Curves ◽  
The Stability ◽  
Area Preserving ◽  
Parabolic Fixed Point

AbstractThe stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.

scholarly journals Parabolic fixed points, invariant curves and action-angle variables

1990 ◽  
Vol 10 (2) ◽  
pp. 231-245 ◽  
Author(s):  
Dov Aharonov ◽  
Uri Elias
Keyword(s):  
Fixed Point ◽  
Hamiltonian System ◽  
Fixed Points ◽  
Phase Flow ◽  
Invariant Curves ◽  
Elliptic Case ◽  
System A ◽  
Twist Mapping ◽  
Area Preserving ◽  
Parabolic Fixed Point

AbstractA fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

The Existence of Analytic Invariant Curves for a Planar Map

2011 ◽  
Vol 225-226 ◽  
pp. 1274-1278
Author(s):  
Ling Xia Liu
Keyword(s):  
Fixed Point ◽  
Two Dimensional ◽  
Complex Field ◽  
Invariant Curves ◽  
Planar Map ◽  
Majorant Series ◽  
Brjuno Condition ◽  
Analytic Invariant

In this paper, we study the existence of analytic invariant curves for two-dimensional maps in the complex field C. Employing the method of majorant series, we discuss the eigenvalueof the mapping at a fixed point. We discuss not only thoseat resonance, i.e., at a root of the unity but also thosenear resonance under Brjuno condition.

Homoclinic and Big Bang Bifurcations of an Embedding of 1D Allee’s Functions into a 2D Diffeomorphism

2017 ◽  
Vol 27 (09) ◽  
pp. 1730030 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Big Bang ◽  
Complete Classification ◽  
Two Dimensional ◽  
Contour Lines ◽  
Dimension One ◽  
Necessary And Sufficient ◽  
Homoclinic Tangencies

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

scholarly journals SIMPLIFIED NUMERICAL FORM OF UNIVERSAL FINITE TYPE INVARIANT OF GAUSS WORDS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350037
Author(s):  
TOMONORI FUKUNAGA ◽  
TAKAYUKI YAMAGUCHI ◽  
TAKAAKI YAMANOI
Keyword(s):  
Normal Form ◽  
Computer Program ◽  
Finite Type ◽  
Group Structure ◽  
Complete Classification ◽  
Finite Type Invariants ◽  
Partial Classification ◽  
Type Invariant

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

Rational maps of ℙn with prescribed fixed points and the smooth conic case

2014 ◽  
Vol 13 (08) ◽  
pp. 1450066 ◽  
Author(s):  
J. A. Vargas ◽  
A. S. Argáez
Keyword(s):  
Fixed Point ◽  
Complete Classification ◽  
Rational Maps ◽  
Plane Curves ◽  
Canonical Forms ◽  
Fixed Point Set ◽  
Constant Degree ◽  
Point Set ◽  
Quadratic Hypersurfaces

We construct rational maps of ℙn which have a prescribed variety as a component of their fixed point set. The resulting maps fix a pencil of lines for the case of hypersurfaces; thus including the cases of plane curves. We also determine the Cremona maps among the constructed ones for quadratic hypersurfaces. Our methods are based on associated matrices of forms of constant degree and the "triple action" of G = PGL n+1 on them. We include a complete classification of these maps and matrices for the case of the smooth conic curve in ℙ2. We obtain invariants and canonical forms for the orbits of our matrices under the triple action of G, modulo syzygies of a row vector. We obtain invariants and canonical forms for the orbits of the constructed rational maps under conjugation by G.

scholarly journals Real entropy rigidity under quasi-conformal deformations

2021 ◽  
Vol 25 (1) ◽  
pp. 1-33
Author(s):  
Khashayar Filom
Keyword(s):  
Complete Classification ◽  
Rational Maps ◽  
Rigidity Result ◽  
Content Type ◽  
The Real ◽  
Real Analytic ◽  
Set Up ◽  
Conformal Deformations ◽  
Classification Of Maps

We set up a real entropy function h R h_\Bbb {R} on the space M d ′ \mathcal {M}’_d of Möbius conjugacy classes of real rational maps of degree d d by assigning to each class the real entropy of a representative f ∈ R ( z ) f\in \Bbb {R}(z) ; namely, the topological entropy of its restriction f ↾ R ^ f\restriction _{\hat {\Bbb {R}}} to the real circle. We prove a rigidity result stating that h R h_\Bbb {R} is locally constant on the subspace determined by real maps quasi-conformally conjugate to f f . As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree d d real maps of real entropy log ⁡ ( d ) \log (d) . The latter discussion moreover entails a complete classification of maps of maximal real entropy.

scholarly journals An Implementation of Lipschitz Simple Functions in Computer Algebra System Singular

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Yanan Liu ◽  
Muhammad Ahsan Binyamin ◽  
Adnan Aslam ◽  
Minahal Arshad ◽  
Chengmei Fan ◽  
...  
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Complete Classification ◽  
Simple Function ◽  
Complex Numbers ◽  
Lipschitz Equivalence

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers ℂ was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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