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Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.

We have generalized the Baker, Devaney and Romera's work and constructed a series of generalized Mandelbort-Julia Sets (in abbreviated form generalized M-J sets) from the complex exponential map. Using the experimental mathematics method, we have innovated as follows: Present the theoretic proof about the explosion of the generalized J sets for complex index number; Theoretically analyze the symmetry and periodicity of the generalized M-J sets; Present a new attaching rule described the distributing of periodicity petal of generalized M sets for complex index number; Find abundant structure information of the generalized J sets contained in the generalized M sets for complex index number; Find that the speed of fractal growth in generalized M-J sets for complex index number is faster than that of generalized M-J sets for real index number, the parameter value @l"0 decides the speed of the fractal growth and the fractal growth in generalized M sets for complex index number points tends to the multifurcation and Misiurewicz point.

Let R be a rational map with a totally disconnected Julia set J(R). If the postcritical set on J(R) contains a non-persistently recurrent (or conical) point, then we show that the map R cannot be a structurally stable map. Introduction and statements Fatou’s problem of the density of hyperbolic maps in the space of rational maps is one of the principal problems in the field of holomorphic dynamics. Due to Mane, Sad and Sullivan [MSS], we can reformulate this problem in the following way: If the Julia set J(R) contains a critical point, then the rational map R is a structurally unstable map. For convenience we give the definition of the structural stability of a rational map. For other basic notations and definitions we refer to the book of Milnor [M]. Definition 1. Let Ratd be the space of all rational maps of degree d with the topology of coefficient convergence. A map R ∈ Ratd is called structurally stable if there exists a neighborhood U ⊂ Ratd of R such that: For any map R1 ∈ U there exists a quasiconformal map f : C → C conjugating R to R1. We give a condition, Assumptions “G” (see below), on a rational map with totally disconnected Julia set and with a critical point on J(R) to be unstable. In the pioneer paper [BH], Branner and Hubbard prove that the Lebesgue measure of the Julia set is zero if there exists only one critical point on J(R). Our result (see Theorem A below) restricted to the Branner–Hubbard case is weaker, but it can be applied for maps with two or more critical points on J(R). Let R be a rational map with a totally disconnected Julia set. Let us normalize R so that the point z = ∞ becomes the attractive fixed point. Let Pc(R) be a postcritical set of the map R and P (R) = Pc(R) ∩ J(R) be a postcritical set on the Julia set. Let S = C\ ⋃ n R −n(Pc(R)), then R : S → S is an unbranched autocovering. Definition 2. We say that a closed simple geodesic γ ⊂ S is linked with P (R) if the interior I(γ) of γ intersects P (R). Received by the editors June 13, 2005. 2000 Mathematics Subject Classification. Primary 37F45; Secondary 37F30. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication

We study the topology of the Julia set of a quadratic Cremer polynomial P . Our main tool is the following topological result. Let f : U → U be a homeomorphism of a plane domain U and let T ⊂ U be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x , then T contains a non-repelling fixed point. Given P , two angles θ , γ are K-equivalent if for some angles x 0 = θ , … , x n = γ the impressions of x i − 1 and x i are non-disjoint, 1 ⩽ i ⩽ n ; a class of K -equivalence is called a K-class . We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense G δ -set of angles each of which is a K -class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K -equivalent.

Let ƒ(z = e2πiθz(1 + z/d)d, θ ∈ ℝ \ ℚ be a polynomial. If θ is an irrational number of bounded type, it is easy to see that ƒ(z has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of ƒ(z satisfies Dim(J(ƒ)) < 2.

In this paper the theory of Julia sets of Schroder iteration functions is introduced, the Julia sets of the Schroder functions of a one-parameter family polynomials with high degree are constructed through iteration method, and their structures are analyzed. Consequently, the following results are found in the study: (1) the Julia sets of the Schroder iteration functions of a one-parameter family polynomials with high degree contain the structure of classical Mandelbrot-like set; (2) the orbits of the critical points may escape from the zero points of the corresponding polynomial to converge to the k-cycle attractive basin or the extra fixed points; (3) if critical points on parameter plane are selected to construct Julia sets on dynamics plane, then attractive k-cycle basin will emerge, while it will not emerge if no critical points are selected; (4) the extra fixed points may be repulsive, litmusless or attractive, but the former takes the major role and (5) the Julia sets of the Schroder iteration functions have symmetry.

Considering the Julia sets J ( T λ ) concerning diamond-like hierarchical Potts models on the complex plane, we proved that T λ is geometrically finite and H D J ( T λ ) > 1 for λ ∈ R .

Recently it was shown in [P. Kim, Fatou's theorem for censored stable processes, Stochastic Process. Appl. 108 (1) (2003) 63–92] that Fatou's theorem for transient censored α-stable processes in a bounded C1,1 open set is true. Here we give a probabilistic proof of relative Fatou's theorem for (−Δ)α/2-harmonic functions (equivalently for symmetric α-stable processes) in bounded κ-fat open set where α∈(0,2). That is, if u is positive (−Δ)α/2-harmonic function in a bounded κ-fat open set D and h is singular positive (−Δ)α/2-harmonic function in D, then nontangential limits of u/h exist almost everywhere with respect to the Martin-representing measure of h. This extends the result of Bogdan and Dyda [K. Bogdan, B. Dyda, Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain, Studia Math. 157 (1) (2003) 83–96]. It is also shown that, under the gaugeability assumption, relative Fatou's theorem is true for operators obtained from the generator of the killed α-stable process in bounded κ-fat open set D through nonlocal Feynman–Kac transforms. As an application, relative Fatou's theorem for relativistic stable processes is also true if D is bounded C1,1-open set.

We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree d ⩾ 2 to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors. We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable.

We consider the dynamics of a transcendental meromorphic function f(z) with only finitely many poles and prove that if f has only finitely many weakly repelling fixed points, then there is no multiply-connected wandering domain in its Fatou set.

In this paper, we analyze the Julia set of Newton method for multiple roots, construct the Julia sets of standard, relaxed and multiple roots. Through the experimental mathematics method we drew the following conclusion: (1) the Julia sets of above methods for f ( z ) = z α ( z β − 1) has β time rotation symmetry and its center is origin; (2) the multiple roots domain of attraction of these kinds of Julia sets are sensitive to α ; (3) there is not simple root domain of attraction in relax method, because z *, the root of f ( z ), is neutral or repelling fixed point of F ( z ); (4) ∞ is not the fixed point of F ( z ), so multiple root’s Julia set consists of a great number of attraction domain of multiple and simple roots; (5) the experimental errors and truncation of coefficients cause the minimal effect to multiple root, the more to relax, the maximal to standard Newton’ method.

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of $\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel capture} type (then $f\in \mathcal{W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of \emph{queer} type (then at least one critical point of $f\in \mathcal{W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and carries an invariant line field)., Comment: 12 pages; one figure; to appear in Proc. Amer. Math. Soc. arXiv admin note: substantial text overlap with arXiv:1305.5799

We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets., Comment: Updated version, to appear in JAMS

We establish several new properties of the escaping setI(f)={z∶f n (z)→∞ andf n(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)⊂J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.

It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set. R. Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point. We provide here the first examples of rational maps whose coefficients are algebraic over $\Q_p$ and that have a (wild) recurrent critical point. In fact, we show that there is such a rational map in every one parameter family of rational maps that is defined over a finite extension of $\Q_p$ and that has a Misiurewicz bifurcation.

The continuity of the map φ : c → dim H(Jc) where Jc is the Julia set of the map fc(z) = z2 + c is discussed in this paper. Using the Hausdorff dimension, a partial order is defined on the Mandelbrot set M. Some properties of irrational points defined by this order are also studied.

(Julia sets of p -adic polynomials and periodic points) An open problem in the iteration theory of p -adic rational functions R is the existence, when the Julia set J ( R ) is not empty, of an repulsive periodic point for R . In this paper, we give a partial positive answer to this question, in the case of a polynomial P , and with some conditions on P .

Meromorphic solutions of the Schroder equation f(sz) = R(f(z)) are studied, where |s| > 1 and R(w) is a rational function with deg[R] ≥ 2. We will show that, if arg[s] /∈ 2πQ, then f(z) has any direction as Borel, and besides, without exceptional values other than Picard values, which depend on R(w). Further the case arg[s] ∈ 2πQ is also considered. We investigate therelation between Julia directions of f(z) and the Julia set of R(w).

We describe how polynomials with all cycles repelling and connected Julia set are organized in the boundary of the shift loci according to their real laminations.

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.

It is shown that for any meromorphic function the Julia set has constant local upper and lower box dimensions, and respectively, near all points of with at most two exceptions. Further, the packing dimension of the Julia set is equal to . Using this result it is shown that, for any transcendental entire function in the class (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of are equal to 2. The approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.

Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of non-trivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers.In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z↦z2+c, where |c

In this paper we prove that hyperbolic Julia sets are locally computable in polynomial time. Namely, for each complex hyperbolic polynomial p(z), there is a Turing machine Mp(z) that can “draw” the set with the precision 2−n, such that it takes time polynomial in n to decide whether to draw each pixel. In formal terms, it takes time polynomial in n to decide for a point x whether d(x,Jp(z))2⋅2−n (in which case we don't draw this pixel). In the case 2−n≤d(x,Jp(x))≤2⋅2−n either answer will be acceptable. This definition of complexity for sets is equivalent to the definition introduced in Weihrauch's book [Weihrauch, K., “Computable Analysis”, Springer, Berlin, 2000] and used by Rettinger and Weihrauch in [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA].Although the hyperbolic Julia sets were shown to be recursive, complexity bounds were proven only for a restricted case in [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA]. Our paper is a significant generalization of [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA], in which polynomial time computability was shown for a special kind of hyperbolic polynomials, namely, polynomials of the form p(z)=z2+c with |c

In this paper we prove the existence of a new type of Sierpinski curve Julia set for certain families of rational maps of the complex plane. In these families, the complementary domains consist of open sets that are preimages of the basin at $\infty$ as well as preimages of other basins of attracting cycles.

In his 84 preprint W. Thurston proved that quadratic laminations do not admit so-called wandering triangles and asked a deep question concerning their existence for laminations of higher degrees. Recently it has been discovered by L. Oversteegen and the author that some closed laminations of the unit circle invariant under $z\mapsto z^d, d>2$ admit wandering triangles. This makes the problem of describing the criteria for the existence of wandering triangles important because solving this problem would help understand the combinatorial structure of the family of all polynomials of the appropriate degree. In this paper for a closed lamination on the unit circle invariant under $z\mapsto z^3$ (cubic lamination) we prove that if it has a wandering triangle then there must be two distinct recurrent critical points in the corresponding quotient space ("topological Julia set") $J$ with the same limit set coinciding with the limit set of any wandering vertex (wandering vertices in $J$ correspond to wandering gaps in the lamination).

In this paper we consider families of rational maps of degree 2n on the Riemann sphereF λ: $$F_\lambda :\bar {\mathbb{C}} \to \bar {\mathbb{C}}$$ given by $$F_\lambda (z) = z^n + \frac{\lambda }{{z^n }}$$ where $$\lambda \in \mathbb{C} - \{ 0\} $$ andn>-2. One of our goals in this paper is to describe a type of structure that we call aCantor necklace that occurs in both the dynamical and the parameter plance forF λ. Roughly speaking, such a set is homeomorphic to a set constructed as follows. Start with the Cantor middle thirds set embedded on thex-axis in the plane. Then replace each removed open interval with an open circular disk whose diameter is the same as the length of the removed interval. A Cantor necklace is a set that is homeomorphic to the resulting union of the Cantor set and the adjoined open disks. The second goal of this paper is to use the Cantor necklaces in the parameter plane to prove the existence of several new types of Sierpinski curve Julia sets that arise in these families of rational maps. Unlike most examples of this type of Julia set, the maps on these Julia sets are structurally unstable. That is, small changes in the parameter λ give rise to Julia sets on which the dynamical behavior is quite different. In addition, we also describe a new type of related Julia set which we call a hybrid Sierpinski curve.

The k-polynomial of a simplicial complex C is the function k C (x)=∑ i⩾1 f i x i where fi is the number of i-faces in C . These k-polynomials are closed under composition, and we are lead to ask: for higher composites of a complex C with itself, what happens to the roots of their k-polynomials? We prove that they converge to the Julia set of k C (x) , thereby associating with C a fractal. For 2-dimensional complexes we exploit the Mandelbrot set to determine when their fractals are connected, and determine the connectness of the fractals for certain families of ‘stripped’ complexes.

A class of rational functions and sufficient conditions on it such that the Julia set is a Jordan arc, respectively no Jordan arc, are introduced. Further, it is shown that any rational function which has a Jordan arc as Julia set belongs up to Mobius conjugation to this class.

The main purpose of the Fatou-Julia theory is to study the global behaviour of the sequence (fn) of iterates of a rational function f. In this survey article we consider generalized iteration which means that the iterated function f may vary from step to step. More precisely, let (fn) be a sequence of rational functions, and let \( F_{n}: = f_{n}\circ \cdots \circ f_{1}\) be the sequence of forward compositions, and let the Fatou set and Julia set of (Fn) be defined as usual. Then, in general, most of the results of the Fatou-Julia theory fail to hold. On the other hand, under appropriate restrictions on the sequence (fn) many results can be carried over to this more general situation, but the proofs are often completely different.

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in \N$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.

Let R be a rational function, of degree ≥2, with complex coefficients. Then the Julia set of R is a closed subset of ℙ1(ℂ), and therefore compact. If one replace ℂ by the field ℂ p (completion of an algebraic closure of the field ℚ p of the p-adic numbers), then one can define also a Julia set for a rational function with p-adic coefficients. But as ℂ p is not locally compact, the Julia set may or may not be compact. In this paper, we study the compactness of the Julia set of p-adic polynomials.

Let 0 << 1 be an irrational number with continued fraction expan- sion =( a1 ;a 2 ;a 3 ;::: ), and consider the quadratic polynomial P : z 7! e 2i z +z 2 . By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log an = O( p n )a sn !1 ; then the Julia set of P is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 << 1, the quadratic P has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P.B y standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.

We show that there is a topological conjugacy between the action of the Hecke group Gn on the completed positive real axis and the action of the shift on base n-1 expressions of real numbers in the unit interval. This conjugacy is shown to occur in holomorphic dynamical systems: we construct (n-1:n-1) holomorphic correspondences (multivalued self-maps of the Riemann sphere) which are matings between Gn acting on the complex upper half plane, and polynomials of degree n-1 acting on their filled Julia sets. Certain of these correspondences split into unions of (2:2) and (1:1) correspondences: we present combinatorial descriptions of limit sets and the connectivity locus in the case n=4.

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.

We consider maps in the tangent family for which the asymptotic values are eventually mapped onto poles. For such functions the Julia set J(f) = C. We prove that for almost all z ∈ J(f) the limit set w(z) is the post-singular set and f is non-ergodic on J(f). We also prove that for such f does not exist a f-invariant measure absolutely continuous with respect to the Lebesgue measure finite on compact subsets of C.

We study the structure of disconnected polynomial Julia sets. We consider polynomials with an arbitrary number of non-escaping critical points, of arbitrary multiplicity, which interact non-trivially. We use a combinatorial system of a tree with dynamics to give a sufficient condition for the Julia set a polynomial to be an area zero Cantor set. We show that there exist uncountably many combinatorially inequivalent polynomials, which satisfy this condition and have multiple non-escaping critical points, each of which accumulates at all the non-escaping critical points.

Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.

In this note, it is shown that if a rational function f of degree ≥2 has a nonempty set of buried points, then for a generic choice of the point z in the Julia set, z is a buried point, and if the Julia set is disconnected, it has uncountably many buried components.

In this note we prove that when P is a polynomial of degree d with connected Julia set and when z 0 belongs to the filled-in Julia set K(P), then |(P'(z 0 )| ≤ d 2 . We also show that equality is achieved if and only if K(P) is a segment of which one extremity is z 0 . In that case, P is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.

For any prime p >0, we contruct p-adic polynomial functions in C p [z] whose Fatou sets have wandering domains. To cite this article: R.L. Benedetto, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 615–620.

Let P be a polynomial whose Julia set J is locally connected. Then a non-preperiodic non-precritical vertex of J must have the limit set which coincides with the limit set of an appropriately chosen recurrent critical point of P. In particular, if all critical points of P are non-recurrent then all vertices of J are preperiodic or precritical.

We call a rational map f graph critical if any critical point either belongs to an invariant finite graph G, or has minimal limit set, or is non-recurrent and has limit set disjoint from G. We prove that, for any conformal measure, either for almost every point of the Julia set J(f) its limit set coincides with J(f), or for almost every point of J(f) its limit set coincides with the limit set of a critical point of f.

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.

Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs. We give an axiomatic definition of self-similar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved. Functional equations and a decomposition algorithm for the Green functions of self-similar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function $d$ on finite directed subgraphs. If the Julia set $\cj$ of $d$ is a Cantor set, then the reciprocal spectrum $\spec^{-1}P=\{1/z\mid z\in\spec P\}$ of the Markov transition operator $P$ can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where $\cd$ is a countable set of the $d$-backwards iterates of a certain finite set of real numbers., Comment: 18 pages, 2 figures, other comments

A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials, even conjectural models are missing, one possible reason being that the higher degree analog of the MLC conjecture is known to be false. We provide a combinatorial model for an essential part of the parameter space of complex cubic polynomials, namely, for the space of all cubic polynomials with connected Julia sets all of whose cycles are repelling (we call such polynomials \emph{dendritic}). The description of the model turns out to be very similar to that of Thurston., Comment: 52 pages, 12 figures (in the new version a few typos have been corrected and some proofs have been expanded). arXiv admin note: substantial text overlap with arXiv:1401.5123

International audience; We study the family of renormalization transformations of the generalized d -dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where d⩾2. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.