scholarly journals Analytic invariants associated with a parabolic fixed point in 2

2008 ◽  
Vol 28 (06) ◽  
pp. 1815 ◽  
Author(s):  
V. GELFREICH ◽  
V. NAUDOT
Keyword(s):  
Fixed Point ◽  
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.

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 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 Cubic polynomials with a parabolic point

2010 ◽  
Vol 30 (6) ◽  
pp. 1843-1867 ◽  
Author(s):  
P. ROESCH
Keyword(s):  
Fixed Point ◽  
Jordan Curve ◽  
Julia Set ◽  
Basin Of Attraction ◽  
Local Connectivity ◽  
Parabolic Point ◽  
Jordan Curves ◽  
Cubic Polynomials ◽  
Parabolic Fixed Point

AbstractWe 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.

Equidistribution of parabolic fixed points in the limit set of Kleinian groups

1999 ◽  
Vol 19 (6) ◽  
pp. 1437-1484 ◽  
Author(s):  
SALVATORE COSENTINO
Keyword(s):  
Fixed Point ◽  
Riemann Hypothesis ◽  
Asymptotic Estimate ◽  
Counting Function ◽  
Weak Limit ◽  
Kleinian Groups ◽  
Limit Set ◽  
Geometrically Finite ◽  
Finite Kleinian Group ◽  
Parabolic Fixed Point

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

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