We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition. What we discuss is the `backward iterated function systems', and since the generator maps are not injective in general, we need some efforts to estimate the measure of disks.
Let G be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that G contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which contains at least three points and is completely invariant under each element of G, is the sphere itself.
We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that generate a non nearly Abelian polynomial semigroup with the property that the Julia set of one generator is equal to the Julia set of the semigroup. These examples show that certain conjectures in the field of dynamics of rational semigroups do not hold as stated and therefore require the allowance of certain exceptional cases.
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of ce…
In holomorphic semigroup dynamics, Julia set is in general backward invariant and so some fundamental results of classical complex dynamics can not be generalized to semigroup dynamics. In this paper, we define completely invariant Julia set of transcendental semigroup and we see how far the results of classical transcendental dynamics generalized to transcendental semigroup dynamics.
We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit functions and postsingular set are also derived. In addition, classes of hyperbolic and postsingularly bounded transcendental semigroups are given. We also provide certain criterion for non existence of wandering domains of transcendental semigroups.
We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We investigate no wandering domain theorem in such a transcendental semigroup. We also obtain results on a complete generalization of the classification of periodic Fatou components.
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely generated rational semigroups (of a certain type). We also provide some consequences of this result.
In this paper, we prove that escaping set of transcendental semigroup is S-forward invariant. We also prove that if holomorphic semigroup is abelian, then Fatou set, Julia set and escaping set are S-completely invariant. We see certain cases and conditions that the holomorphic semigroup dynamics exhibits same dynamical behavior just like the classical complex dynamics. Frequently, we also examine certain amount of connection and contrast between classical complex dynamics and holomorphic semigroup dynamics.
We define commutator of a transcendental semigroup, and on the basis of this concept, we define conjugate semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given semigroup is nearly abelian. We also prove that image of each of escaping set, Julia set and Fatou set under commutator (affine complex conjugating maps) is equal to respectively escaping set, Julia set and Fatou set of conjugate semigroup. Finally, we prove that every element of the nearly abelian semigroup $ S $ can be written as the composition of an element from the set generated by the set of c…
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the…
We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup $G$ generated by a compact family of rational maps on the Riemann sphere of degree two or greater, the Julia set of any subsemigroup of $G$ is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is sem…
We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $Γ$, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $γ\in Γ^{\Bbb{N}}$, the Julia set $J_γ$ of $γ$ is a Jordan curve but not a quasicircle, t…
We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatatio…
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [David Fried, Sebastian. M. Marotta, and Rich Stankewitz, Complex dynamics of Möbius semigroups, Ergodic Theory Dynam. Systems, 32(6):1889--1929, 2012].
We investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup $ S $ are non-empty.
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that …
We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen's formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite …
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. Moreover, we show that for any $n\in \Bbb{N} \cup \{\aleph_{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all co…
We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic, i.e., for which the critical point $0$ is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric $ρ(z) |dz|$, where $ρ(z) = 1+\frac{1}{\textrm{dist}(z,P(f))^{1-1/d}}$, and where $P(f)$ is the postcritical set of $f$. We also show that this metric is Hölder equivalent to the usual Euclidean metric.
We consider the multifractal formalism for the dynamics of semigroups of rational maps on the Riemann sphere and random complex dynamical systems. We elaborate a multifractal analysis of level sets given by quotients of Birkhoff sums with respect to the skew product associated with a semigroup of rational maps. Applying these results, we perform a multifractal analysis of the Hölder regularity of limit state functions of random complex dynamical systems.
We investigate the dynamics of $2$-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space ${\cal B}$ of such semigroups. We show that for a parameter $h$ in the intersection of ${\cal B}$, the hyperbolicity locus ${\cal H}$ and the closure of the disconnectedness locus (the space of parameters for which the Julia set is disconnected), the corresponding semigroup satisfies either the open set condition (and the Bowen's formula) or that the Julia sets of the two generators coincide. Also, we sh…
This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typ…
We show that the Julia set of a non-elementary rational semigroup G is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of G. This also proves that the limit set of a non-elementary Moebius group is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of the group and this implies that the limit set of a finitely generated non-elementary Kleinian group is uniformly perfect.
Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were introduced shortly after rational semigroups as analogues in higher real dimensions, but have received far less attention. Each map in a quasiregular semigroup must necessarily be a uniformly quasiregular map. While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them. In this paper,…
We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic fam…
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $F$ and the Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$. Moreover, we show that the equality holds if and only if the generators $f_{j}$ are simultaneously conjugate to the form $a_{j}z^{\pm…
We consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroup case. We also investigate the dynamics of conjugate semigroups, Abelian transcendental semigroups and wandering and Baker domains of transcendental semigroups.
In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. We then use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacit…
In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to ana…
We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and prove the existence of stationary probability measures.
We consider expanding semiflows on branched surfaces. The family of transfer operators associated to the semiflow is a one-parameter semigroup of operators. The transfer operators may also be viewed as an operator-valued function of time and so, in the appropriate norm, we may consider the vector-valued Laplace transform of this function. We obtain a spectral result on these operators and relate this to the spectrum of the generator of this semigroup. Issues of strong continuity of the semigroup are avoided. The main result is the improvement to the machinery associated with studying semiflows…
We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex variable complex/holomorphic dynamics) and the theory of countable alphabet conformal iterated function systems (CIFSs). We construct the thermodynamic formalism for such dynamical systems and geometric potentials by developing the notion of nice families that extend to the case of our highly disconnected skew product phase space the powerful notion of nice se…
We introduce the concept of escaping set for semigroups of transcendental entire functions using Fatou-Julia theory. Several results of the escaping set associated with the iteration of one transcendental entire function have been extended to transcendental semigroups. We also investigate the properties of escaping sets for conjugate semigroups and abelian transcendental semigroups. Several classes of transcendental semigroups for which Eremenko's conjectures hold have been provided.
We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical co…
Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalise these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prov…
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
We investigate the random dynamics of polynomial maps on the Riemann sphere and the dynamics of semigroups of polynomial maps on the Riemann sphere. In particular, the dynamics of a semigroup $G$ of polynomials whose planar postcritical set is bounded and the associated random dynamics are studied. In general, the Julia set of such a $G$ may be disconnected. We show that if $G$ is such a semigroup, then regarding the associated random dynamics, the chaos of the averaged system disappears in the $C^{0}$ sense, and the function $T_{\infty}$ of probability of tending to $\infty$ is Hölder continu…
The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure $m$ with minimal exponent $h$; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a $\sg$--finite invariant measure $μ$ …
In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet
We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+α}$ conformal) is robustly minimal (resp. ergodic with respect to Lebesgue measure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.
We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, "singular functions on the complex plane" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity wit…
We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou components are Jordan disks with disjoint closures (Sierpiński maps), as well as those where any two Fatou components can be connected through a countable chain of Fatou components with common boundary points (crochet or Newton-like maps). We provide several alternative characterizations for our decomposition, as well as an algorithm for its effective computation. …
Let $Γ$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $Γ$ on $\R^d$ is strongly irreducible and that $Γ$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $Γ$ on $\R^d$ at infinity. This amounts to the consideration of the action of $Γ$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space $\pr^{d-1}.$ In the case where $Γ$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $Γ$ has the above properties, we deduce that the $Γ$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.
