scholarly journals Domain of Attraction for Maps Tangent to the Identity in $$\mathbb {C} ^2$$ C 2 with Characteristic Direction of Higher Degree

2015 ◽  
Vol 26 (4) ◽  
pp. 2519-2541
Author(s):  
Sara Lapan
Keyword(s):  
Domain Of Attraction ◽  
Characteristic Direction ◽  
Tangent To The Identity

scholarly journals ATTRACTING DOMAINS OF MAPS TANGENT TO THE IDENTITY WHOSE ONLY CHARACTERISTIC DIRECTION IS NON-DEGENERATE

2013 ◽  
Vol 24 (10) ◽  
pp. 1350083 ◽  
Author(s):  
SARA W. LAPAN
Keyword(s):  
Domain Of Attraction ◽  
Existence Results ◽  
Characteristic Direction ◽  
Tangent To The Identity

Let f be a holomorphic germ on ℂ2 that fixes the origin and is tangent to the identity. Assume that f has a non-degenerate characteristic direction [v]. Hakim gave conditions that guarantee the existence of attracting domains along [v], however, when f has only one characteristic direction, these conditions are not satisfied. We prove that when [v] is unique, the existence results still hold. In particular, there is a domain Ω whose points converge to the origin along [v] and, on Ω, f is conjugate to a translation. Furthermore, if f is a global automorphism, the corresponding domain of attraction is a Fatou–Bieberbach domain.

scholarly journals Characteristic directions of two-dimensional biholomorphisms

2020 ◽  
Vol 156 (5) ◽  
pp. 869-880
Author(s):  
Lorena López-Hernanz ◽  
Rudy Rosas
Keyword(s):  
Fixed Points ◽  
Two Dimensional ◽  
Characteristic Direction ◽  
Parabolic Curve ◽  
Analytic Curve ◽  
Tangent To The Identity

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

THE NON-DICRITICAL ORDER AND ATTRACTING DOMAINS OF HOLOMORPHIC MAPS TANGENT TO THE IDENTITY

2014 ◽  
Vol 25 (01) ◽  
pp. 1450003 ◽  
Author(s):  
FENG RONG
Keyword(s):  
Fixed Point ◽  
Duke Math ◽  
Holomorphic Maps ◽  
Characteristic Direction ◽  
Local Dynamics ◽  
Analytic Transformation ◽  
Tangent To The Identity

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp, 0) tangent to the identity, Duke Math. J.92 (1998) 403–428], n - 1 invariants αj, 1 ≤ j ≤ n - 1, called the directors, were associated to [v] and it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Re αj = 0 for some j. With the help of a new invariant μ called the non-dicritical order, we show that f has an attracting domain at p tangent to [v] if μ ≥ 1. We also study the "spiral domains" when μ = 0. For n = 2, we show that f has an attracting domain at p tangent to [v] if and only if either the director α > 0 or μ ≥ 1.

scholarly journals MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

2019 ◽  
Vol 35 (6) ◽  
pp. 1234-1270 ◽  
Author(s):  
Sébastien Fries ◽  
Jean-Michel Zakoian
Keyword(s):  
Conditional Distribution ◽  
Stable Distribution ◽  
Financial Time Series ◽  
Domain Of Attraction ◽  
Real Data ◽  
Autoregressive Processes ◽  
Financial Time ◽  
Causal Representation ◽  
Heavy Tailed ◽  
Non Gaussian

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

Regional stabilization and H∞ congestion control with input saturation

2021 ◽  
pp. 014233122199273
Author(s):  
Sadek Belamfedel Alaoui ◽  
El Houssaine Tissir ◽  
Noreddine Chaibi ◽  
Fatima El Haoussi
Keyword(s):  
Linear Systems ◽  
Controller Design ◽  
Input Saturation ◽  
Domain Of Attraction ◽  
Queue Management ◽  
Variable Delay ◽  
Challenging Problem ◽  
Regional Stabilization ◽  
Small Gain ◽  
Improved Performance

Designing robust active queue management subjected to network imperfections is a challenging problem. Motivated by this topic, we addressed the problem of controller design for linear systems with variable delay and unsymmetrical constraints by the scaled small gain theorem. We designed two mechanisms: robust enhanced proportional derivative; and robust enhanced proportional derivative subjected to input saturation. Discussion of their practical implementations along with extensive comparisons by MATLAB and NS3 illustrate the improved performance and the enlargement of the domain of attraction regarding some literature results.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

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