ordinal rankings on measures annihilating thin sets we assign a countable ordinal number to each probability measure which annihilates all h sets the descriptive set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non borel in addition it allows us to quantify the failure of rajchman s conjecture similar results are obtained for measures annihilating dirichlet sets

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

strong convergence theorems by hybrid methods for families of nonexpansive mappings in hilbert spaces abstract in this paper we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes nakajo and takahashi s theorems k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 simultaneously furthermore we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from nakajo and takahashi using this theorem we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a hilbert space

equidistribution towards the bifurcation current i multipliers and degree d polynomials in the moduli space mathcal p d of degree d polynomials the set text per n w of classes f for which f admits a cycle of exact period n and multiplier multiplier w is known to be an algebraic hypersurface we prove that given w in mathbb c these hypersurfaces equidistribute towards the bifurcation current as n tends to infinity

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

rational misiurewicz maps are rare we show that the set of misiurewicz maps has lebesgue measure zero in the space of rational functions for any fixed degree d 2

bifurcation currents in holomorphic dynamics on p k potential theory has been introduced in one dimensional rational dynamics by brolin and tortrat 4 29 but does not play a central role there in higher dimension however as the classical tools are not any longer efficient pluri potential theory has revealed itself to be essential the fundamental works of hubbard papadopol fornaess sibony bedfordsmillie briend duval see 26 for precise references enlight the remarkable efficency of pluri potential theory in holomorphic dynamics on pk or ck it is therefore tempting to study the parameter spaces in a similar way more precisely one would like to relate the bifurcations of an holomorphic family f x of endomorphisms of pk to the powers of a certain current on the parameter space x let us recall that in dimension k 1 a bifurcation is said to occur at some point 0 x if the julia set of f does not move continuously around 0 the famous work of mane sadsullivan 16 which is based on the lemma and the fatou cremer sullivan classification relates the bifurcations with the instability of the critical orbits it also establishes that the bifurcations concentrate on the complement of an open dense subset of x for the quadratic family z2 x c the bifurcation locus is precisely the boundary of the mandelbrot set

weak and strong convergence theorems for strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the mann s algorithm generates a sequence x n by the formula x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then x n converges weakly to a fixed point of t however this convergence is in general not strong we then modify mann s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence this result extends a recent result of nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 from nonexpansive mappings to strict pseudo contractions

monotone cq iteration processes for nonexpansive semigroups and maximal monotone operators abstract nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 proved strong convergence theorems for nonexpansive mappings nonexpansive semigroups and the proximal point algorithm for zero point of monotone operators in hilbert spaces by the cq iteration method the purpose of this paper is to modify the cq iteration method of k nakajo and w takahashi using the monotone cq method and to prove strong convergence theorems in the proof process of this article the cauchy sequence method is used so we proceed without use of the demiclosedness principle and opial s condition and other weak topological techniques

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

random walks kleinian groups and bifurcation currents let be a holomorphic family of representations of a finitely generated group g into psl 2 parameterized by a complex manifold we define a notion of bifurcation current in this context that is a positive closed current on describing the bifurcations of this family of representations in a quantitative sense it is the analogue of the bifurcation current introduced by demarco for holomorphic families of rational mappings on 1 our definition relies on the theory of random products of matrices so it depends on the choice of a probability measure on g

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

equivalence and topological invariance of conditions for non uniform hyperbolicity in the iteration of rational maps we show equivalence of several standard conditions for non uniform hyperbolicity of complex rational functions including the topological collet eckmann condition tce uniform hyperbolicity on periodic orbits exponential shrinking of components of pre images of small discs backward collet eckmann condition at one point positivity of the infimum of lyapunov exponents of finite invariant measures on the julia set the condition tce is stated in purely topological terms so we conclude that all these conditions are invariant under topological conjugacy for rational maps with one critical point in julia set all the conditions above are equivalent to the usual collet eckmann and backward collet eckmann conditions thus the latter ones are invariant by topological conjugacy in the unicritical setting we also prove that neither part of this stronger statement is valid in the multicritical case

approximation of non archimedean lyapunov exponents and applications over global fields let k be an algebraically closed field of characteristic 0 that is complete with respect to a non archimedean absolute value we establish a locally uniform approximation formula of the lyapunov exponent of a rational map f of mathbb p 1 of degree d 1 over k in terms of the multipliers of n periodic points of f with an explicit control in terms of n f and k as an immediate consequence we obtain an estimate for the blow up of the lyapunov exponent near a pole in one dimensional families of rational maps over k combined with our former archimedean version this non archimedean quantitative approximation allows us to show a quantified version of silverman s and ingram s recent comparison between the critical height and any ample height on the moduli space mathcal m d bar mathbb q two improvements of mcmullen s finiteness of the multiplier maps reduction to multipliers of cycles of exact given period and an effective bound from below on the period a characterization of non affine isotrivial rational maps defined over the function field mathbb c x of a normal projective variety x in terms of the growth of the degree of the multipliers

growth of balls of holomorphic sections and energy at equilibrium let l be a big line bundle on a compact complex manifold x given a non pluripolar compact subset k of x and a continuous hermitian metric e on l we define the energy at equilibrium of k as the monge ampere energy of the extremal psh weight associated to k we prove the differentiability of the energy at equilibrium with respect to and we show that this energy describes the asymptotic behaviour as k of the volume of the sup norm unit ball induced by k k on the space of global holomorphic sections h 0 x kl as a consequence of these results we recover and extend rumely s robin type formula for the transfinite diameter we also obtain an asymptotic description of the analytic torsion and extend yuan s equidistribution theorem for algebraic points of small height to the case of a big line bundle

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

an implicit iteration process for nonexpansive semigroups let c be a closed convex subset of a banach space e let t t t 0 be a strongly continuous semigroup of nonexpansive mappings on c such that t 0f t t 0 let n and tn be sequences of real numbers satisfying appropriate conditions then for arbitrary x0 c the mann type implicit iteration process xn given by xn nxn 1 1 n t tn xn n 0 weakly strongly converges to an element of t 0f t t

the collet eckmann condition for rational functions on the riemann sphere we show that the set of collet eckmann maps has positive lebesgue measure in the space of rational maps on the riemann sphere for any fixed degree d 2

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

dynamics of horizontal like maps in higher dimension we study the regularity of the green currents and of the equilibrium measure associated to a horizontal like map in under a natural assumption on the dynamical degrees we estimate the speed of convergence towards the green currents the decay of correlations for the equilibrium measure and the lyapounov exponents we show in particular that the equilibrium measure is hyperbolic we also show that the green currents are the unique invariant vertical and horizontal positive closed currents the results apply in particular to henon like maps to regular polynomial automorphisms of and to their small perturbations

distribution of rational maps with a preperiodic critical point let f lambda be a family of rational maps of a fixed degree with a marked critical point c lambda under a natural assumption we first prove that the hypersurfaces of parameters for which c lambda is periodic converge as a sequence of positive closed 1 1 currents to the bifurcation current attached to c and defined by demarco we then turn our attention to the parameter space of polynomials of a fixed degree d by intersecting the d 1 currents attached to each critical point of a polynomial bassaneli and berteloot obtained a positive measure mu rm bif of finite mass which is supported on the connectedness locus they showed that its support is included in the closure of the set of parameters admitting d 1 neutral cycles we show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials i e of misiurewicz points

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

lyapunov exponents bifurcation currents and laminations in bifurcation loci bifurcation loci in the moduli space of degree d rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period n and multiplier 0 or ei using potential theoretic arguments we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current to this purpose we first establish approximation formulas for the lyapunov function in degree d 2 this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin exists

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

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

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

complex projective structures lyapunov exponent degree and harmonic measure we study several new invariants associated to a holomorphic projective struc ture on a riemann surface of finite analytic type the lyapunov exponent of its holonomy which is of probabilistic dynamical nature and was introduced in our previous work the degree which measures the asymptotic covering rate of the developing map and a family of harmonic measures on the riemann sphere previously introduced by hussenot we show that the degree and the lyapunov exponent are related by a simple formula and give estimates for the hausdorff dimension of the harmonic measures in terms of the lyapunov exponent in accordance with the famous sullivan dictionary this leads to a descrip tion of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics

viscosity approximation methods for pseudo contractive semigroups in banach spaces abstract we study the strong convergence of two viscosity iteration processes for pseudo contractive semigroup and for strongly pseudo contractive mapping in uniformly convex banach spaces with uniformly g teaux differentiable norm as special cases we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain banach spaces the results presented in this paper extend and generalize previous results

the measure of non normal sets it is well known that almost every number in 0 1 is normal in base 2 in the sense of lebesgue measure kahane and salem asked whether the same is true with respect to any borel measure whose fourier stieltjes coefficients vanish at infinity in other words whether the set of non normal numbers is a set of uniqueness in the wide sense we show that this is not the case in fact we give best possible conditions on the rate of decay of hat mu n in order that almost every number be normal the techniques include on the one hand probability measures with respect to which the binary digits in 0 1 are independent only by blocks rather than individually and on the other hand the strong law of large numbers for weakly correlated random variables

strong convergence theorem for nonexpansive semigroups in hilbert space abstract the purpose of this paper is to prove the strong convergence of a method combining the descent method and the hybrid method in mathematical programming for finding a point in the common fixed point set of a semigroup of nonexpansive mappings in hilbert space

positive lyapunov exponents in families of one dimensional dynamical systems we give a generalization of jakobson s theorem ja and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures

fine structure of connectedness loci we study distributions of the harmonic and lebesgue measures on the boundary of the connectedness locus mathcal m d for uni critical polynomials z d c and dynamical properties which are typical with respect to the harmonic measure one result is a conformal similarity between mathcal m d and the corresponding julia set for almost every point of mathcal m d with respect to the harmonic measure it is also shown that the harmonic measure is supported on lebesgue density points of the complement of mathcal m d which are not accessible from outside within john angles and at which the boundary of mathcal m d spirals infinitely often in both directions in the case of quadratic polynomials we prove that every parameter c from the boundary of the mandelbrot set mathcal m that is recurrent under iterates of z 2 c but not infinitely tunable is a lebesgue weak density point of the complement of mathcal m a direct consequence of our theory is the yoccoz local connectivity theorem a new technical ingredient in the paper is the method of amplification which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding julia set the pointwise conformality of the similarity map is derived through inducing techniques and a classical twb theory

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

on the support of the bifurcation measure of cubic polynomials we construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by misiurewicz polynomials in particular such parameters admit maximal bifurcations but do not belong to the support of the bifurcation measure

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

dynamics of rational maps lyapunov exponents bifurcations and capacity let l f log df d f denote the lyapunov exponent of a rational map f p 1 p 1 in this paper we show that for any holomorphic family of rational maps f x of degree d 1 t f dd c l f defines a natural positive 1 1 current on x supported exactly on the bifurcation locus of the family the proof is based on the following potential theoretic formula for the lyapunov exponent here f c 2 c 2 is a homogeneous polynomial lift of f g f is the escape rate function of f and capk f is the homogeneous capacity of the filled julia set of f we show in particular that the capacity of k f is given explicitly by the formula where res f is the resultant of the polynomial coordinate functions of f we introduce the homogeneous capacity of compact circled and pseudoconvex sets k c 2 and show that the levi measure determined by the geometry of k is the unique equilibrium measure such k c 2 correspond to metrics of non negative curvature on p 1 and we obtain a variational characterization of curvature

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

on the stability of iterative approximations to fixed points of nonexpansive mappings abstract we study iterative retraction approximations to fixed points of the nonexpansive self mapping a g g given on the closed convex set g in a banach space b the conditions which guarantee weak and strong convergence and stability of these approximations with respect to perturbations of both operator a and constraint set g are considered the results of this paper are new even in a hilbert space for the iterative projection approximations

super potentials of positive closed currents intersection theory and dynamics we introduce a notion of super potential for positive closed currents of bidegree p p on projective spaces this gives a calculus on positive closed currents of arbitrary bidegree we define in particular the intersection of such currents and the pull back operator by meromorphic maps one of the main tools is the introduction of structural discs in the space of positive closed currents which gives a geometry on that space we apply the theory of super potentials to construct green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms

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

misiurewicz points in one dimensional quadratic maps abstract misiurewicz points are constituted by the set of unstable or repellent points sometimes called the set of exceptional points these points which are preperiodic and eventually periodic play an important role in the ordering of hyperbolic components of one dimensional quadratic maps in this work we use graphic tools to analyse these points by measuring their preperiods and periods and by ordering and classifying them

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

misiurewicz parameters and dynamical stability of polynomial like maps of large topological degree given a family of polynomial like maps of large topological degree we relate the presence of misiurewicz parameters to a growth condition for the volume of the iterates of the critical set this generalizes to higher dimensions the well known equivalence between stability and normality of the critical orbits in dimension one we also introduce a notion of holomorphic motion of asymptotically all repelling cycles and prove its equivalence with other notions of stability our results allow us to generalize the theory of stability and bifurcation developed by berteloot dupont and the author for the family of all endomorphisms of mathbb p k of a given degree to any arbitrary family of endomorphisms of mathbb p k or polynomial like maps of large topological degree

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

hybrid mann halpern iteration methods for nonexpansive mappings and semigroups abstract in this paper we introduce some new iteration methods based on the hybrid method in mathematical programming the mann s iterative method and the halpern s method for finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpansive semigroup hilbert spaces