scholarly journals Superattracting fixed points of quasiregular mappings

2014 ◽  
Vol 36 (3) ◽  
pp. 781-793 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Fixed Point ◽  
Spherical Shell ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Basin Of Attraction ◽  
Quasiregular Mapping ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Polynomial Type ◽  
The Basin Of Attraction

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

scholarly journals Strategies for an Efficient Official Publicity Campaign

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Juan Neirotti
Keyword(s):  
Fixed Point ◽  
Public Opinion ◽  
Phase Space ◽  
Fixed Points ◽  
Basin Of Attraction ◽  
Opinion Formation ◽  
Publicity Campaign ◽  
The Basin Of Attraction

AbstractWe consider the process of opinion formation, in a society where there is a set of rules B that indicates whether a social instance is acceptable. Public opinion is formed by the integration of the voters’ attitudes which can be either conservative (mostly in agreement with B) or liberal (mostly in disagreement with B and in agreement with peer voters). These attitudes are represented by stable fixed points in the phase space of the system. In this article we study the properties of a perturbative term, mimicking the effects of a publicity campaign, that pushes the system from the basin of attraction of the liberal fixed point into the basin of the conservative point, when both fixed points are equally likely.

scholarly journals Hausdorff measure of the singular set of quasiregular maps on Carnot groups

2003 ◽  
Vol 2003 (35) ◽  
pp. 2203-2220 ◽  
Author(s):  
Irina Markina
Keyword(s):  
Vector Space ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Quasiregular Mapping ◽  
Carnot Groups ◽  
Singular Set ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Dimensional Hausdorff Measure ◽  
Homogeneous Dimension

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Letνstand for the homogeneous dimension of a Carnot group and letmbe the index of the last vector space of the corresponding Lie algebra. We prove that the(ν−m−1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimates of the local index of quasiregular mappings are also obtained.

Biaccessibility in quadratic Julia sets

2000 ◽  
Vol 20 (6) ◽  
pp. 1859-1883 ◽  
Author(s):  
SAEED ZAKERI
Keyword(s):  
Fixed Point ◽  
Measure Zero ◽  
Julia Set ◽  
Basin Of Attraction ◽  
Countable Union ◽  
Common Theme ◽  
Chebyshev Map ◽  
Straight Line ◽  
The Common ◽  
The Basin Of Attraction

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.

The effects of parameter variation on the gaits of passive walking models: simulations and experiments

Robotica ◽  
2009 ◽  
Vol 27 (4) ◽  
pp. 511-528 ◽  
Author(s):  
Liu Ning ◽  
Li Junfeng ◽  
Wang Tianshu
Keyword(s):  
Fixed Points ◽  
Basin Of Attraction ◽  
Step Length ◽  
Moment Of Inertia ◽  
Mass Centre ◽  
Random Slope ◽  
Simulation Results ◽  
Good For ◽  
The Times ◽  
The Basin Of Attraction

SUMMARYWe have made a systematic study of the gait of a straight leg planar passive walking model through simulations and experiments. Three normalised parameters, which represent the foot radius, the position of the mass centre and the moment of inertia, are used to characterise the walking model.In the simulation, we have obtained the fixed points and the basins of attraction of the walking models with different parameter combinations by the aid of the cell mapping method. With the results of fixed points, we investigated the effects of parameter variations on the gait descriptors, including step length, period, average speed and energy inefficiency. A model that has a large basin of attraction has been obtained, and it can start walking far from its fixed point. However, the size of the basin of attraction is not a good measurement of robustness. Thus, we proposed floors with random slope angles or stairs with random heights to test robustness. Five hundred times of simulations with 100 non-dimensional time units were implemented for each parameter combination. The times that the walker failed to arrive at the end were recorded. The simulation results showed that the model with a larger foot radius and higher position of mass centre has a lower possibility of falling on uneven floors. A large moment of inertia is helpful for walking on a random slope angle floor, while low values of moment of inertia are good for navigating random stairs.Prototype experiments have validated the simulation results, which showed that models with larger feet have a longer step length and high speed. However, period differences were difficult to obtain in the experiments since the differences were very small. We have tested the sensitivity with the initial conditions of the models with different foot radii on a flat floor, and have also tested the robustness of the models on a floor with random slope angles. The times that the model reached the end of the floor were recorded. The experimental results showed that a large foot radius is good for improving the basin of attraction and robustness on uneven floors. Finally, the exceptions of the experiment are explained.

scholarly journals Capture Zones of the Family of Functions λzmexp(z)

2003 ◽  
Vol 13 (09) ◽  
pp. 2623-2640 ◽  
Author(s):  
Núria Fagella ◽  
Antonio Garijo
Keyword(s):  
Fixed Point ◽  
Critical Point ◽  
Basin Of Attraction ◽  
Capture Zone ◽  
Connected Component ◽  
Dynamical Plane ◽  
Capture Zones ◽  
The Family ◽  
Parameter Values ◽  
The Basin Of Attraction

We consider the family of entire transcendental maps given by Fλ,m(z)=λzm exp (z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

scholarly journals Effective Uniform Bounds on the Krasnoselski-Mann Iteration

2000 ◽  
Vol 7 (9) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Convex Sets ◽  
Fixed Point Theory ◽  
Uniformly Convex ◽  
Asymptotic Regularity ◽  
Normed Spaces ◽  
Mann Iteration ◽  
Starting Point

This paper is a case study in proof mining applied to non-effective proofs<br />in nonlinear functional analysis. More specifically, we are concerned with the<br />fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex<br />spaces the rate of convergence is in general not computable in f (which is<br />related to the non-uniqueness of fixed points). However, the iterations yield<br />approximate fixed points of arbitrary quality for general normed spaces and<br />bounded C (asymptotic regularity).<br />In this paper we apply general proof theoretic results obtained in previous<br />papers to non-effective proofs of this regularity and extract uniform explicit<br />bounds on the rate of the asymptotic regularity. We start off with the classical<br />case of uniformly convex spaces treated already by Krasnoselski and show<br />how a logically motivated modification allows to obtain an improved bound. Already the analysis of the original proof (from 1955) yields an elementary<br />proof for a result which was obtained only in 1990 with the use of the deep<br />Browder-G¨ohde-Kirk fixed point theorem. The improved bound from the modified<br /> proof gives applied to various special spaces results which previously had<br />been obtained only by ad hoc calculations and which in some case are known<br />to be optimal.<br />The main section of the paper deals with the general case of arbitrary normed<br />spaces and yields new results including a quantitative analysis of a theorem<br />due to Borwein, Reich and Shafrir (1992) on the asymptotic behaviour of<br />the general Krasnoselski-Mann iteration in arbitrary normed spaces even for unbounded sets C. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform<br />bounds which do not depend on the starting point of the iteration and the<br />nonexpansive function and the normed space X and, in fact, only depend<br />on the error epsilon, an upper bound on the diameter of C and some very general information on the sequence of scalars k used in the iteration. Even non-effectively only the existence of bounds satisfying weaker uniformity conditions was known before except for the special situation, where lambda_k := lambda is constant. For the unbounded case, no quantitative information was known so far.

Accessible saddles on fractal basin boundaries

1992 ◽  
Vol 12 (3) ◽  
pp. 377-400 ◽  
Author(s):  
Kathleen T. Alligood ◽  
James A. Yorke
Keyword(s):  
Fixed Point ◽  
Basin Of Attraction ◽  
Open Disk ◽  
Fractal Basin Boundaries ◽  
Basin Boundary ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Simply Connected ◽  
Basin Boundaries ◽  
The Basin Of Attraction

AbstractFor a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.

Entropy Production of the Willamowski-Rössler Oscillator

2005 ◽  
Vol 60 (8-9) ◽  
pp. 599-9 ◽  
Author(s):  
Jörg W. Stucki ◽  
Robert Urbanczik
Keyword(s):  
Fixed Point ◽  
Entropy Production ◽  
Dynamic Behaviour ◽  
Basin Of Attraction ◽  
Steady States ◽  
Original Model ◽  
Core Component ◽  
Mean Values ◽  
The Mean ◽  
The Basin Of Attraction

Some properties of the Willamowski-Rössler model are studied by numerical simulations. From the original equations a minimal version of the model is derived which also exhibits the characteristic properties of the original model. This minimal model shows that it contains the Volterra-Lotka oscillator as a core component. It thus belongs to a class of generalized Volterra-Lotka systems. It has two steady states, a saddle point, responsible for chaos, and a fixed point, dictating its dynamic behaviour. The chaotic attractor is located close to the surface of the basin of attraction of the saddle node. The mean values of the variables are equal to the (unstable) steady state values during oscillations even under chaos, and the variables are always non-negative as in other generalized Volterra-Lotka systems. Surprisingly this was also the case with the original reversible Willamowski-Rössler model allowing to compare the entropy production during oscillations with the entropy production of the steady states. During oscillations the entropy production was always lower even under chaos. Since under these circumstances less energy is dissipated to produce the same output, the oscillating system is more efficient than the non-oscillatory one.

Punctured Parabolic Cylinders in Automorphisms of ℂ2

2020 ◽  
Author(s):  
Josias Reppekus
Keyword(s):  
Fixed Point ◽  
Blow Up ◽  
Basin Of Attraction ◽  
Irrational Rotation ◽  
Parabolic Cylinder ◽  
Density Property ◽  
Fatou Component ◽  
Crucial Step ◽  
Blowing Up ◽  
The Basin Of Attraction

Abstract We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.

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