scholarly journals The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

2007 ◽  
Vol 345 (12) ◽  
pp. 695-698 ◽  
Author(s):  
Colin Christopher ◽  
Christiane Rousseau
Keyword(s):  
Fixed Point ◽  
Moduli Space ◽  
Parabolic Fixed Point

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.

Parabolic Fixed Points of Kleinian Groups and the Horospherical Foliation on Hyperbolic Manifolds

1997 ◽  
Vol 08 (02) ◽  
pp. 289-299 ◽  
Author(s):  
A. N. Starkov
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Limit Point ◽  
Elementary Proof ◽  
Hyperbolic Manifolds ◽  
Kleinian Groups ◽  
Complicated Structure ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Parabolic Fixed Point

We use dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso (ℍn). Let ℋ be the horospherical foliation on the unit tangent bundle SM of manifold M = Γ\ℍn with constant negative curvature. We construct examples Γ ⊂ Iso (ℍ4) which show that horosphere based at parabolic fixed point w ∈ ∂ℍ4 can project to leaf ℋx ⊂ SM of complicated structure: it can be locally closed and not closed; not locally closed and non-dense in the non-wandering set Ω+ ⊂ SM of ℋ; dense in Ω+ (this is equivalent to w being a horospherical limit point). Using the natural duality, one gets the corresponding examples of Γ-orbits on the light cone. We give an elementary proof of the fact that conical limit point w ∈ ∂ℍn cannot be a parabolic fixed point.

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.

Parabolic-like mappings

2014 ◽  
Vol 35 (7) ◽  
pp. 2171-2197 ◽  
Author(s):  
LUNA LOMONACO
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
The Family ◽  
Dynamical Properties ◽  
Image Position ◽  
Parabolic Fixed Point

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

scholarly journals EQUATIONS OF THE MODULI OF HIGGS PAIRS AND INFINITE GRASSMANNIAN

2009 ◽  
Vol 20 (08) ◽  
pp. 1029-1055 ◽  
Author(s):  
D. HERNÁNDEZ-SERRANO ◽  
J. M. MUÑOZ PORRAS ◽  
F. J. PLAZA MARTÍN
Keyword(s):  
Fixed Point ◽  
Moduli Space ◽  
Vector Bundles ◽  
Higgs Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Spectral Cover ◽  
Relative Version ◽  
Moduli Of Vector Bundles

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

scholarly journals SPECIAL CURVES AND POSTCRITICALLY FINITE POLYNOMIALS

2013 ◽  
Vol 1 ◽  
Author(s):  
MATTHEW BAKER ◽  
LAURA DE MARCO
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Moduli Space ◽  
Dense Subset ◽  
Rational Curves ◽  
Rational Maps ◽  
Complex Polynomial ◽  
Polynomial Dynamical Systems ◽  
Postcritically Finite ◽  
Cubic Polynomials

AbstractWe study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

Linearization of holomorphic germs with quasi-parabolic fixed points

2008 ◽  
Vol 28 (3) ◽  
pp. 979-986 ◽  
Author(s):  
FENG RONG
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Analytic Functions ◽  
Classical Result ◽  
Linear Part ◽  
Analytic Form ◽  
Moscow Math ◽  
Parabolic Fixed Point

AbstractLet f be a germ of a holomorphic diffeomorphism of $\mathbb {C}^n$ with the origin O being a quasi-parabolic fixed point, i.e. the spectrum of dfO consists of 1 and e2iπθj with $\theta _j\in \mathbb {R}\!\setminus \!\mathbb {Q}$. We show that f is locally holomorphically conjugated to its linear part, if f is of some particular form and its eigenvalues satisfy certain arithmetic conditions. When the spectrum of dfO does not consist of any 1’s, this is the classical result of Siegel [C. L. Siegel. Iteration of analytic functions. Ann. of Math.43 (1942), 607–612] and Brjuno [A. D. Brjuno. Analytic form of differential equations. Trans. Moscow Math. Soc.25 (1971), 131–288; 26 (1972), 199–239].

scholarly journals Comptage des -chtoucas: la partie elliptique

2013 ◽  
Vol 149 (12) ◽  
pp. 2169-2183 ◽  
Author(s):  
Ngo Dac Tuan
Keyword(s):  
Fixed Point ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Shimura Varieties

AbstractWe extend our previous work in collaboration with Ngô Bao Châu and give a fixed point formula for the elliptic part of moduli spaces of$G$-shtukas with arbitrary modifications. Our formula is similar to the fixed point formula of Kottwitz for certain Shimura varieties. Our method is inspired by that of Kottwitz and simpler than that of Lafforgue for the fixed point formula of the moduli space of Drinfeld$\text{GL} (r)$-shtukas.

Sign in / Sign up

Export Citation Format

Share Document