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

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

differential embedding of the lorenz attractor ideally an embedding of an n dimensional dynamical system is n dimensional ideally an embedding of a dynamical system with symmetry is symmetric ideally the symmetry of the embedding is the same as the symmetry of the original system this ideal often cannot be achieved differential embeddings of the lorenz system which possesses a twofold rotation symmetry are not ideal while the differential embedding technique happens to yield an embedding of the lorenz attractor in three dimensions it does not yield an embedding of the entire flow an embedding of the flow requires at least four dimensions the four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in r4 this inversion symmetric three manifold cannot be projected into any three dimensional euclidean subspace without singularities

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

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

twisted orlicz algebras and complete isomorphism to operator algebras abstract let g be a locally compact group let g g c be a 2 cocycle and let denote the twisted convolution associated to given a complementary pair of young functions we prove that if is continuous and strictly increasing l g l 2 g and 0 1 s t u s v t s t g for some u v s g then l g with the maximal operator space structure is completely isomorphic to an operator algebra we also present various classes of 2 cocycles for which one could obtain such algebras generalizing in part the results of 15 we apply our methods to compactly generated groups of polynomial growth and demonstrate that our results could be applied to variety of cases

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

covering dynamical systems twofold covers we study the relation between a dynamical system which is unchanged equivariant under a discrete symmetry group g and another locally identical dynamical system with no residual symmetry we also study the converse mapping lifting a dynamical system without symmetry to a multiple cover which is equivariant under g this is done in r 3 for the two element rotation and inversion groups comparisons are done for the equations of motion the strange attractors that they generate and the branched manifolds that classify these strange attractors a dynamical system can have many inequivalent multiple covers all equivariant under the same symmetry group g these are distinguished by the value of a certain topological index many examples are presented a new global bifurcation the peeling bifurcation is described

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

polynomials without repelling periodic point of given period abstract bergweiler proved that for any given integer k 2 every polynomial p of degree d 2 has at least one repelling periodic cycle of period k unless k d 2 2 2 3 2 4 3 2 here we classified these exceptional polynomials we also showed that the julia sets of these exceptional polynomials are connected

comparison of tests for embeddings it is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test the classical tests which depend on real numbers fractal dimensions lyapunov exponents averaged over an attractor are compared with a topological test that depends on integers the comparison can only be done for mappings into three dimensions we find that the classical tests fail to predict when a mapping is an embedding and when it is not we point out the reasons for this failure which are not restricted to three dimensions

newton maps as matings of cubic polynomials in this paper we prove existence and uniqueness of matings of a large class of renormalizable cubic polynomials with one fixed critical point and the other cubic polynomial having two fixed critical points the resulting mating is a newton map our result is the first part toward a conjecture by l tan stating that all cubic newton maps can be described as matings or captures

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

preperiodic points and unlikely intersections in this article we combine complex analytic and arithmetic tools to study the preperiodic points of one dimensional complex dynamical systems we show that for any fixed a b c and any integer d 2 the set of c c for which both a and b are preperiodic for z c is infinite if and only if a b this provides an affirmative answer to a question of zannier which itself arose from questions of masser concerning simultaneous torsion sections on families of elliptic curves using similar techniques we prove that if rational functions c z have infinitely many preperiodic points in common then all of their preperiodic points coincide and in particular the maps must have the same julia set this generalizes a theorem of mimar who established the same result assuming that and are defined over q the main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields with non archimedean berkovich spaces playing an essential role

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

example of a weighted algebra lpw g on an uncountable discrete group abstract we construct examples of weighted algebras l p w g with 1 p 2 on uncountable free groups for p 2 no weighted algebras exist on these groups on the other hand we prove that for any amenable locally compact group g if l p w g is an algebra for some weight w and p 1 then g is compact

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

combinatorics and topology of straightening maps i compactness and bijectivity abstract we study the parameter space structure of degree d 3 one complex variable polynomials as dynamical systems acting on c we introduce and study straightening maps these maps are a natural higher degree generalization of the ones introduced by douady and hubbard to prove the existence of small copies of the mandelbrot set inside itself we establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems also we characterize straightening maps with compact domains moreover we give two classes of bijective straightening maps the first produces an infinite collection of embedded copies of the d 1 fold product of the mandelbrot set in the connectedness locus of degree d 3 the second produces an infinite collection of full families of quadratic connected filled julia sets in the cubic connectedness locus such that each filled julia set is quasiconformally embedded

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

computation of domains of analyticity for some perturbative expansions of mechanics abstract we compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics we use pade approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence we also use non perturbative methods based on direct computation of the invariant circles and in analogy with greene s criterion approximation by circles with rational rotation we find that the domains computed by all the methods agree within the limits of accuracy we also study rigorously the nature of the singularities when the frequency is rational

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

all schatten spaces endowed with the schur product are q algebras we prove that the banach algebra formed by the space of compact operators on a hilbert space endowed with the schur product is a quotient of a uniform algebra also known as a q algebra together with a similar result of perez garcia for the trace class this completes the answer to a long standing question of varopoulos

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

non landing parameter rays of the multicorns it is well known that every rational parameter ray of the mandelbrot set lands at a single parameter we study the rational parameter rays of the multicorn mathscr m d the connectedness locus of unicritical antiholomorphic polynomials of degree d and give a complete description of their accumulation properties one of the principal results is that the parameter rays accumulating on the boundaries of odd period except period 1 hyperbolic components of the multicorns do not land but accumulate on arcs of positive length consisting of parabolic parameters we also show the existence of undecorated real analytic arcs on the boundaries of the multicorns which implies that the centers of hyperbolic components do not accumulate on the entire boundary of mathscr m d and the misiurewicz parameters are not dense on the boundary of mathscr m d

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

antipode preserving cubic maps the fjord theorem this note will study a family of cubic rational maps which carry antipodal points of the riemann sphere to antipodal points we focus particularly on the fjords which are part of the central hyperbolic component but stretch out to infinity these serve to decompose the parameter plane into subsets each of which is characterized by a corresponding rotation number

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

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

embeddings of low dimensional strange attractors topological invariants and degrees of freedom when a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding dependent we show that there are just three topological properties that depend on the embedding parity global torsion and knot type we discuss how they can change with the embedding finally we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings these results apply only to chaotic attractors of genus one which covers the majority of cases in which experimental data have been subjected to topological analysis this means that the conclusions drawn from previous analyses for example that the mechanism generating chaotic behavior is a smale horseshoe mechanism a reverse horseshoe a gateau roul e an s template branched manifold etc are not artifacts of the embedding chosen for the analysis

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

global topology from an embedding an embedding of chaotic data into a suitable phase space creates a diffeomorphism of the original attractor with the reconstructed attractor although diffeomorphic the original and reconstructed attractors may not be topologically equivalent in a previous work we showed how the original and reconstructed attractors can differ when the original is three dimensional and of genus one type in the present work we extend this result to three dimensional attractors of arbitrary genus this result describes symmetries exhibited by the lorenz attractor and its reconstructions

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

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

representation theory for strange attractors embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image different embeddings may or may not be equivalent under isotopy we regard embeddings as representations of the attractor review the labels required to distinguish inequivalent representations for an important class of dynamical systems and discuss the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one universal embedding in a suitable dimension

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

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

spurious lyapunov exponents in attractor reconstruction lyapunov exponents perhaps the most informative invariants of a complicated dynamical process are also among the most difficult to determine from experimental data in particular when using embedding theory to build chaotic attractors in a reconstruction space extra open quotes spurious close quotes lyapunov exponents arise that are not lyapunov exponents of the original system the origin of these spurious exponents is discussed and formulas for their determination in the low noise limit are given copyright ital 1998 ital the american physical society

when are projections also embeddings we study an autonomous four dimensional dynamical system used to model certain geophysical processes this system generates a chaotic attractor that is strongly contracting with four lyapunov exponents ensuremath lambda i that satisfy ensuremath lambda 1 ensuremath lambda 2 ensuremath lambda 3 l0 so the lyapunov dimension is d l 2 ensuremath mid ensuremath lambda 3 ensuremath mid ensuremath lambda 1 l3 in the range of coupling parameter values studied as a result it should be possible to find three dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior we study mappings into r 3 to determine which can be used as embeddings to reconstruct the dynamics we find dramatically different behavior in the two simplest mappings projections from r 4 to r 3 in one case the one parameter family of attractors studied remains topologically unchanged for all coupling parameter values in the other case during an intermediate range of parameter values the projection undergoes self intersections while the embedded attractors at the two ends of this range are topologically mirror images of each other