a lower bound on the canonical height for polynomials we prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit the lower bound depends only on the degree of the polynomial the degree of the number field and the number of places of bad reduction

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

local connectivity of julia sets for unicritical polynomials we prove that the julia set j f of at most finitely renormalizable unicritical polynomial f z d c with all periodic points repelling is locally connected for d 2 it was proved by yoccoz around 1990 it follows from a priori bounds in a modified principal nest of puzzle pieces the proof of a priori bounds makes use of new analytic tools developed in kl09 that give control of moduli of annuli under maps of high degree

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

herman s condition and siegel disks of bi critical polynomials we extend a theorem of herman from the case of unicritical polynomials to the case of polynomials with two finite critical values this theorem states that siegel disks of such polynomials under a diophantine condition called herman s condition on the rotation number must have a critical point on their boundaries

critical heights on the moduli space of polynomials abstract let m d be the moduli space of one dimensional degree d 2 complex polynomial dynamical systems the escape rates of the critical points determine a critical heights map g m d r d 1 for generic values of g we show that each connected component of a fiber of g is the deformation space for twist deformations on the basin of infinity we analyze the quotient space t d obtained by collapsing each connected component of a fiber of g to a point the space t d is a parameter space analog of the polynomial tree t f associated to a polynomial f c c studied in demarco and mcmullen 2008 6 and there is a natural projection from t d to the space of trees t d we show that the projectivization p t d is compact and contractible further the shift locus in p t d has a canonical locally finite simplicial structure the top dimensional simplices are in one to one correspondence with topological conjugacy classes of structurally stable polynomials in the shift locus

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

total nonpositivity of nonsingular matrices in this paper nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns these characterizations extend an existing characterization that uses some restrictive hypotheses

moduli space of cubic newton maps abstract in this article we study the topology and bifurcations of the moduli space m 3 of cubic newton maps it s a subspace of the moduli space of cubic rational maps carrying the riemann orbifold structure c 2 3 we prove two results the boundary of the unique unbounded hyperbolic component is a jordan arc and the boundaries of all other hyperbolic components are jordan curves the head s angle map is surjective and monotone the fibers of this map are characterized completely the first result is a moduli space analogue of the first author s dynamical regularity theorem 37 the second result confirms a conjecture of tan lei

infinitely renormalizable quadratic polynomials we prove that the julia set of a quadratic polynomial which admits an infinite sequence of unbranched simple renormalizations with complex bounds is locally connected the method in this study is three dimensional puzzles

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

a characterization of hyperbolic rational maps we give a topological characterization of rational maps with disconnected julia sets our results extend thurston s characterization of postcritically finite rational maps in place of iteration on teichmuller space we use quasiconformal surgery and thurston s original result

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

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

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

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

inner totally positive matrices abstract suppose a m n is a staircase matrix with row and column staircase sequences 1 2 n 1 2 n a minor a det a with 1 k 1 k is said to be an inner minor of a if i i i i for i 1 2 k a is said to be inner totally positiv e itp if every inner minor of a is positive we prove that a is itp if a 0 for all inner minors with q k n 0 k 1 2 n also a m n is itp iff it is totally non negative and its extreme inner minors are positive we show that an itp matrix may be reduced by similarity transformations to an itp band matrix and may alternatively be filled in by similarity transformations to become a tp matrix in both the reduction and the filling in the matrix is itp at each stage the analysis is applied to some inverse eigenvalue problems for band matrices

sign structure preserves for m banded factorizations of sign regular matrices abstract a square matrix a a ij is called m banded if a ij 0 for all i j m motivated by the results of rainey and habetler j w rainey g j habetler tridiagonalization of completely nonnegative matrices math comp 26 1972 121 128 we provide m banded factorizations of sign regular matrices that preserve their sign structures as applications some eigenvalue interlacing properties and majorization relations are obtained for certain sign regular matrices

sign regular matrices and neville elimination abstract a pivoting strategy of o n operations for the neville elimination of n n nonsingular sign regular matrices is introduced among other nice properties it is proved that it preserves sign regularity it is also shown its relationship with scaled partial pivoting strategies for neville elimination

quadratic like dynamics of cubic polynomials a small perturbation of a quadratic polynomial f with a non repelling fixed point gives a polynomial g with an attracting fixed point and a jordan curve julia set on which g acts like angle doubling however there are cubic polynomials with a non repelling fixed point for which no perturbation results into a polynomial with jordan curve julia set motivated by the study of the closure of the cubic principal hyperbolic domain we describe such polynomials in terms of their quadratic like restrictions

on local connectivity for the julia set of rational maps newton s famous example we show that newton s cubic methods famous rational maps have a locally connected julia set except in some very specific cases in particular when those maps are infinitely renormalizable their julia set is locally connected and contains small copies of nonlocally connected quadratic julia sets this also holds when newton s method is renormalizable and has cremer points unlike the polynomial case after a dynamical description we show the necessity of the brjuno condition within this family

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

rigidity for real polynomials we prove the topological or combinatorial rigidity property for real polynomials with all critical points real and nondegenerate which completes the last step in solving the density of axiom a conjecture in real one dimensional dynamics

frontiers in complex dynamics rational maps on the riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems first rational maps are complexanalytic so a broad spectrum of techniques can contribute to their study quasiconformal mappings potential theory algebraic geometry etc the rational maps of a given degree form a finite dimensional manifold so exploration of this parameter space is especially tractable finally some of the conjectures once proposed for smooth dynamical systems and now known to be false seem to have a definite chance of holding in the arena of rational maps in this article we survey a small constellation of such conjectures centering around the density of hyperbolic rational maps those which are dynamically the best behaved we discuss some of the evidence and logic underlying these conjectures and sketch recent progress towards their resolution

mating non renormalizable quadratic polynomials in this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1 2 limb of mathcal m is non renormalizable and does not have any non repelling periodic orbits

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

the iteration of cubic polynomials part i the global topology of parameter space fonctions univalentes en dynamique analytique complexe torsion de la structure complexe topologie globale de l espace parametre

bounds for maps of an interval with one critical point of inflection type ii recently quite a few papers appeared proving complex bounds local connectivity of julia sets absence of invariant linefields and quasisymmetric rigidity for real polynomial maps let us first discuss the unimodal case in 1990 martens ma proved that there are real bounds for a certain sequence of first return maps to suitable central intervals in 1990 sullivan showed that if the unimodal maps are real analytic infinitely renormalizable and have bounded combinatorics then their first return maps extend to the complex plane as polynomial like maps see su and also ms in 1994 the present authors proved that all real analytic unimodal maps allow such polynomial like complex extensions for the non central first return maps the proof even gave explicit numerical lower bounds for the moduli of certain annuli ls1 for the quadratic case different proofs were later provided by gs1 ly for the smooth case see ko in 1996 sands see sa improved our numerical bounds for maps of the form g x so that the schwarzian derivative of g is negative this polynomial like structure is one of the main ingredients in many recent results on smooth unimodal maps for example it is heavily used in the proof

nonsingular almost strictly sign regular matrices abstract the class of nonsingular almost strictly totally positive matrices has been characterized m gasca j m pena characterizations and decompositions of almost strictly positive matrices siam j matrix anal appl 28 2006 1 8 in this paper we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices a characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns in particular we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors

local connectivity and quasi conformal rigidity of non renormalizable polynomials we prove that topologically conjugate non renormalizable polynomials are quasi conformally conjugate from this we derive that each such polynomial can be approximated by a hyperbolic polynomial as a by product we prove that the julia set of a non renormalizable polynomial with only hyperbolic periodic points is locally connected and the branner hubbard conjecture the main tools are the enhanced nest construction developed in a previous joint paper with rigidity for real polynomials ann of math 2 165 2007 749 841 and a lemma of kahn and lyubich for which we give an elementary proof in the real case

dynamics of mcmullen maps abstract in this article we develop the yoccoz puzzle technique to study a family of rational maps termed mcmullen maps we show that the boundary of the immediate basin of infinity is always a jordan curve if it is connected this gives a positive answer to the question of devaney higher regularity of this boundary is obtained in almost all cases we show that the boundary is a quasi circle if it contains neither a parabolic point nor a recurrent critical point for the whole julia set we show that the mcmullen maps have locally connected julia sets except in some special cases