scholarly journals Loewner evolution of hedgehogs and 2-conformal measures of circle maps

2020 ◽  
pp. 1-20
Author(s):  
KINGSHOOK BISWAS
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Unit Disk ◽  
Infinitesimal Generator ◽  
Acta Math ◽  
Loewner Equation ◽  
Circle Maps ◽  
Loewner Evolution ◽  
The Family ◽  
Conformal Measures

Abstract Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $${\mathbb C}$$ (i.e. $$f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $$(g_t)$$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $$(g_t)$$ is the orbit of a locally defined semigroup $$(\Phi _t)$$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $$(K_t)$$ . We show that the Loewner measures $$(\mu _t)$$ driving the equation are 2-conformal measures on the circle for the circle maps $$(g_t)$$ .

scholarly journals Positive area and inaccessible fixed points for hedgehogs

2015 ◽  
Vol 36 (6) ◽  
pp. 1839-1850 ◽  
Author(s):  
KINGSHOOK BISWAS
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Fixed Points ◽  
Acta Math ◽  
Rigid Rotation ◽  
Circle Maps ◽  
Dimension One

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0)=0,f^{\prime }(0)=e^{2{\it\pi}i{\it\alpha}},{\it\alpha}\in \mathbb{R}-\mathbb{Q}$). Perez-Marco showed the existence of a unique family of non-trivial invariant full continua containing the fixed point called Siegel compacta. When $f$ is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation $R_{{\it\alpha}}(z)=e^{2{\it\pi}i{\it\alpha}}z$) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs [Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294]; these were used by the author to construct hedgehogs of Hausdorff dimension one, and adapted by Cheritat [Relatively compact Siegel disks with non-locally connected boundaries, Preprint, 2009] to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals Stratified, Weak Stratified, and Three-Valued Semantics1

1990 ◽  
Vol 13 (1) ◽  
pp. 19-33
Author(s):  
Melvin Fitting ◽  
Marion Ben-Jacob
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Horn Clause ◽  
Logic Programs ◽  
The Family ◽  
Fixed Point Semantics ◽  
Truth Value ◽  
The Relationship ◽  
Classical Truth

We investigate the relationship between three-valued Kripke/Kleene semantics and stratified semantics for stratifiable logic programs. We first show these are compatible, in the sense that if the three-valued semantics assigns a classical truth value, the stratified approach will assign the same value. Next, the familiar fixed point semantics for pure Horn clause programs gives both smallest and biggest fixed points fundamental roles. We show how to extend this idea to the family of stratifiable logic programs, producing a semantics we call weak stratified. Finally, we show weak stratified semantics coincides exactly with the three-valued approach on stratifiable programs, though the three-valued version is generally applicable, and does not require stratification assumptions.

scholarly journals Strong andΔ-Convergence Theorems for Common Fixed Points of a Finite Family of Multivalued Demicontractive Mappings in CAT(0)Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
C. E. Chidume ◽  
A. U. Bello ◽  
P. Ndambomve
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Common Fixed Points ◽  
Finite Family ◽  
Iterative Sequence ◽  
Convergence Theorems ◽  
The Family ◽  
Demicontractive Mappings ◽  
Nonlinear Mappings

LetKbe a nonempty closed and convex subset of a complete CAT(0) space. LetTi:K→CBK,i=1,2,…,m, be a family of multivalued demicontractive mappings such thatF:=⋂i=1mF(Ti)≠∅. A Krasnoselskii-type iterative sequence is shown toΔ-converge to a common fixed point of the familyTi,i=1,2,…,m. Strong convergence theorems are also proved under some additional conditions. Our theorems complement and extend several recent important results on approximation of fixed points of certain nonlinear mappings in CAT(0)spaces. Furthermore, our method of the proof is of special interest.

Fixed points of composite meromorphic functions and normal families

2004 ◽  
Vol 134 (4) ◽  
pp. 653-660 ◽  
Author(s):  
Walter Bergweiler
Keyword(s):  
Fixed Point ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Normal Families ◽  
The Family

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

Bifurcations of dynamic rays in complex polynomials of degree two

1992 ◽  
Vol 12 (3) ◽  
pp. 401-423 ◽  
Author(s):  
Pau Atela
Keyword(s):  
Fixed Points ◽  
Julia Set ◽  
Period Doubling ◽  
Mandelbrot Set ◽  
Complex Polynomials ◽  
Circle Maps ◽  
External Rays ◽  
The Family ◽  
Parameter Values ◽  
Dynamic Plane

AbstractIn the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M, the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M. We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M, in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

scholarly journals CONFORMAL HOUSE

2010 ◽  
Vol 25 (24) ◽  
pp. 4603-4621 ◽  
Author(s):  
THOMAS A. RYTTOV ◽  
FRANCESCO SANNINO
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Gauge Group ◽  
Gauge Theories ◽  
Simultaneous Presence ◽  
Consistency Check ◽  
Effective Lagrangian ◽  
Anomalous Dimensions ◽  
Mass Terms

We investigate the gauge dynamics of nonsupersymmetric SU (N) gauge theories featuring the simultaneous presence of fermionic matter transforming according to two distinct representations of the underlying gauge group. We bound the regions of flavors and colors which can yield a physical infrared fixed point. As a consistency check we recover the previously investigated bounds of the conformal windows when restricting to a single matter representation. The earlier conformal windows can be imagined to be part now of the new conformal house. We predict the nonperturbative anomalous dimensions at the infrared fixed points. We further investigate the effects of adding mass terms to the condensates on the conformal house chiral dynamics and construct the simplest instanton induced effective Lagrangian terms.

scholarly journals Common fixed points of single-valued and multivalued maps

2005 ◽  
Vol 2005 (19) ◽  
pp. 3045-3055 ◽  
Author(s):  
Yicheng Liu ◽  
Jun Wu ◽  
Zhixiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Common Fixed Points ◽  
Partial Answer ◽  
Multivalued Maps ◽  
Hybrid Pair ◽  
Common Fixed Point Theorems

We define a new property which contains the property (EA) for a hybrid pair of single- and multivalued maps and give some new common fixed point theorems under hybrid contractive conditions. Our results extend previous ones. As an application, we give a partial answer to the problem raised by Singh and Mishra.

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