julia sets and mandelbrot sets in noor orbit in recent literature researchers have generated julia sets and mandelbrot sets in mann and ishikawa orbits that are examples of two step and three step feedback processes respectively this paper presents further generalization of julia and mandelbrot sets for complex valued polynomials such as quadratic cubic and higher degree polynomials using a noor orbit which is a four step iterative procedure the graphical images of julia and mandelbrot sets have been visualized and certain patterns in mandelbrot sets have been recognized it is fascinating to see that a few mandelbrot sets are akin to a butterfly or a coupled urn or a coupled trident

unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

asymptotic equivalence of neutral and infinite retarded differential equations staffan a demontre qu une equation differentielle fonctionnelle fde neutre avec d operateur stable peut souvent etre traitee comme une fde a retard avec retardement infini le but principal de l article est d illustrer comment les techniques de staffans peuvent etre employees pour etudier le comportement asymptotique de solutions de certaines fde neutres non lineaires et non autonomes en particulier on considere une equation neutre scalaire de la forme d dt x t cx t r f t x t x t r

scaling of mandelbrot sets generated by critical point preperiodicity letz f z be a complex holomorphic function depending holomorphically on the complex parameter if for 0 a critical point off0 falls after a finite number of steps onto an unstable fixed point off0 then in the parameter space near 0 an infinity of more and more accurate copies of the mandelbrot set appears we compute their scaling properties

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

topological entropy of quadratic polynomials and dimension of sections of the mandelbrot set abstract a fundamental theme in holomorphic dynamics is that the local geometry of parameter space e g the mandelbrot set near a parameter reflects the geometry of the julia set hence ultimately the dynamical properties of the corresponding dynamical system we establish a new instance of this phenomenon in terms of entropy indeed we prove an entropy formula relating the entropy of a polynomial restricted to its hubbard tree to the hausdorff dimension of the set of rays landing on the corresponding vein in the mandelbrot set the results contribute to the recent program of w thurston of understanding the geometry of the mandelbrot set via the core entropy

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

collet eckmann and the bifurcation measure the moduli space mathcal m d of degree d ge 2 rational maps can naturally be endowed with a measure mu text bif detecting maximal bifurcations called the bifurcation measure we prove that the support of the bifurcation measure mu text bif has positive lebesgue measure to do so we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of mu text bif and we exhibit a large set of collet eckmann rational maps which satisfy this condition as a consequence we get a set of collet eckmann rational maps of positive lebesgue measure which are approximated by hyperbolic rational maps

fixed points of feigenbaum s type for the equationf p λx λf x existence and hyperbolicity of fixed points for the mapn p f x 1f p x withf p p fold iteration and f p 0 are given forp large these fixed points come close to being quadratic functions and our proof consists in controlling perturbation theory about quadratic functions

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

spirals in the mandelbrot set ii abstract the alpha function is used to quantify the asymptotic structure of the various branches and embedded spirals around the left hand side of the main cardioid in the mandelbrot set

concepts of strict positive realness and the absolute stability problem of continuous time systems after the conditions for absolute stability and positive realness are reviewed it is shown that the current definition of positive realness is inadequate to establish less conservative criteria for absolute stability in this paper the concepts of wide positive realness wpr and wide strict positive realness wspr are proposed both wpr and wspr functions may have poles in the open left and right half planes and wpr functions may have single poles on the imaginary axis including j using the concepts of wpr and wspr less conservative criteria including algebraic and frequency domain criteria are presented for absolute stability of slope restricted multiple nonlinearity feedback systems refined results for non slope restricted nonlinearity feedback systems are also proposed it is pointed out that there is no spectral factorization for wpr and wspr functions if they have poles at both j and j this can be used to explain the deficiencies of algebraic criteria presented in previous publications

julia sets in parameter spaces given a complex number of modulus 1 we show that the bifurcation locus of the one parameter family fb z z bz2 z3 b contains quasi conformal copies of the quadratic julia set j z z2 as a corollary we show that when the julia set j z z2 is not locally connected for example when z z z2 has a cremer point at 0 the bifurcation locus is not locally connected to our knowledge this is the first example of complex analytic parameter space of dimension 1 with connected but non locally connected bifurcation locus we also show that the set of complex numbers of modulus 1 for which at least one of the parameter rays has a non trivial accumulation set contains a dense g subset of s1

on quasi conformal in compatibility of satellite copies of the mandelbrot set i douady and hubbard ann sci ec norm suppl 4 18 2 287 343 1985 introduced the notion of polynomial like maps they used it to identify homeomorphic copies mathbf m of the mandelbrot set inside the mandelbrot set mathbf m these copies can be primitive with a root cusp or satellite without a root cusp they conjectured that the primitive copies are quasiconformally homeomorphic to mathbf m and that the satellite ones are quasiconformally homeomorphic to mathbf m outside any small neighbourhood of the root these conjectures are now theorems due to lyubich ann math 149 319 420 1999 the satellite copies mathbf m p q are clearly not q c homeomorphic to mathbf m but are they mutually q c homeomorphic or even q c homeomorphic to half of the logistic mandelbrot set in this paper we prove that in general the induced douady hubbard homeomorphism is not the restriction of a q c homeomorphism for any two satellite copies mathbf m and mathbf m the induced douady hubbard homeomorphism is not q c if the root multipliers lambda text e 2 pi i p q and lambda text e 2 pi i p q have q not q

a stability inequality for nonlinear feedback systems with slope restricted nonlinearity using lyapunov s direct method a stability inequality in frequency domain is derived for nonlinear feedback systems with slope restricted nonlinearity in the present approach a transformed system that involves the slope of the nonlinearity is considered thus leading to a stability inequality that incorporates the slope information but not the sector information of the nonlinearity the approach is essentially an extension of a recent idea of the author

the scenery flow for hyperbolic julia sets we define the scenery flow space at a point z in the julia set j of a hyperbolic rational map t c c with degree at least 2 and more generally for t a conformal mixing repellor we prove that for hyperbolic rational maps except for a few exceptional cases listed below the scenery flow is ergodic we also prove ergodicity for almost all conformal mixing repellors here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real analytic curves and furthermore that for the collection of such maps based on a fixed open set u the ergodic cases form a dense open subset of that collection scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the hausdorff measure hd where d is the dimension of j and that the flow has a unique measure of maximal entropy for all conformal mixing repellors the flow is loosely bernoulli and has topological entropy at most d moreover the flow at almost every point is the same up to a rotation and so as a corollary one has an analogue of the lebesgue density theorem for the fractal set giving a different proof of a theorem of falconer

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique