Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane

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 with respect to the random dynamics of polynomials. Under certain conditions these functions $T$ are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions $T$ are continuous on the Riemann sphere and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the non-differentiability and regularity of these functions. We find many phenomena which can hold in the random complex dynamics and the dynamics of semigroups of rational maps, but cannot hold in the usual iteration dynamics of a single holomorphic map. We carry out a systematic study of these phenomena and their mechanisms.

Let D be the open unit disk in the complex plane, and let C be its boundary, the unit circle.If x £ C, then by an arc at x we mean a simple arc y with one end point at x such that y-{x}£ö.If/is a function mapping D into some metric space M, then the set of curvilinear convergence of/is defined to be {x e C: there exists an arc y at x and there exists a point p e M such that f(z) ->p as z -»■ x along y}.If ^ is a function whose domain is a subset F of the set of curvilinear convergence of/ then <f> is called a boundary function for/if, and only if, for each x e E there exists an arc y at x such that/(z) -*■ <f>(x) as z -> x along y.Let 5 be another metric space.We shall say that a function <f> is of Baire class ^ 1(5, M) if (i) domain <£ = S, (ii) range ^ £ M, and(iii) there exists a sequence {<£"} of continuous functions, each mapping S into M, such that <f>n^>-<p pointwise on 5. We shall say that (f> is of honorary Baire class ^ 2(S, M) if (i) domain <f> = S, (ii) range ^£M, and (iii) there exists a countable set TVs 5 and there exists a function y!r of Baire class ^ 1(5, M) such that <f>(x) = tp(x) for every xe S-N.It is known that if/is a continuous function mapping F into the Riemann sphere, then the set of curvilinear convergence off is of type Fa6, and any boundary function for/is of honorary Baire class ^2(C, Riemann sphere).(See [3], [4], [5], [6],[9].) J. E. McMillan [6] posed the following problem.If A is a given set in C of type Fao, and if </> is a function of honorary Baire class Ú2(A, Riemann sphere), does there always exist a continuous function / mapping D into the Riemann sphere such that A is the set of curvilinear convergence of/ and <f> is a boundary function for /?The purpose of this paper is to give an affirmative answer to McMillan's question.However, the corresponding question for real-valued functions remains open.(See Problems 1 and 2 at the end of this paper.)In proving our result, we first give a proof under the assumption that <j> is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere.As we shall indicate in an appendix, the same device can be

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). 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 ) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.

In Sections 1 and 2 we treat normal families of meromorphic functions. These are families that are sequentially compact when regarded as functions with values in the extended complex plane. We give two characterizations of normal families, Marty’s theorem in Section 1 and the Zalcman lemma in Section 2. From the latter characterization we deduce Montel’s theorem on compactness of families of meromorphic functions that omit three points, and we also prove the Picard theorems. Sections 3 and 4 constitute an introduction to iteration theory and Julia sets. In Section 3 we proceed far enough into the theory to see how Montel’s theorem enters the picture and to indicate the fractal nature (self-similarity) of Julia sets. In Section 4 we relate the connectedness of Julia sets to the orbits of critical points. In Section 5 we introduce the Mandelbrot set, which has been called the “most fascinating and complicated subset of the complex plane.”

This chapter defines the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. Topological polynomials are topological dynamical systems that generalize complex polynomials with locally connected Julia sets restricted to their Julia sets and considered up to topological conjugacy. This chapter aims to illustrate the analogy between the dynamics of topological polynomials on their cutpoints and cut-atoms and interval dynamics. For example, it is known that for interval maps, periodic points and critical points play a significant, if not decisive, role. This chapter thus attempts to establish similar facts for topological polynomials.

The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton's method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence.

1. In our research on approximation by polynomial and rational interpolation in the complex plane we were led to some problems on balayage (sweeping out) which we want to describe. Let D be a bounded simply connected domain in the complex plane C. Let C = C ∪∞ be the extended complex plane and let A be defined by A = C\D, where D = D ∪ ∂D and ∂D is the boundary of D. In the problems below we consider unit (probability) measures α and β such that α(D) = 1 or α(D) = 1, and β(A) = 1 or β(A) = 1. In all cases α′ and β ′ denote the balayage onto ∂D of α and β, respectively. We denote by δ∞ the Dirac measure at infinity, and by δ′ ∞ its balayage onto ∂D. Then δ ′ ∞ is the equilibrium measure for logarithmic potentials of ∂D. We refer to [6] for this and other facts on balayage. In the problems below we are interested in the equality α′ = β ′. In Problem 1 we consider the case β = δ∞.

One of the most studied fractals corresponds to the Julia Set and one of its elements can be obtained with the following recursive formula in the complex plane: Zi+1 = Zi2 - 1, the result depends on the chosen starting complex number. We can define the Julia set of a complex variable polynomial as the boundary of the set of points that escape to infinity when iterating this polynomial. This means that the orbit of an element of the Julia set does not escape infinity.

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a &gt; 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

Let T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ &gt; 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) &gt; supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

A complex valued function defined and analytic in the entire complex plane is called an entire function or an integral function. In other words, in the extended complex plane, it can have a singularity only at infinity. If f is an entire function different from a polynomial and M(r, f) [or M(r) when there is no confusion] denotes the maximum of |f(z)| on the circle |z| = r, it follows from Liou-ville’s theorem that M(r, f) increases more rapidly than any power of r. Thus, to study the growth of the function we have to compare with exponentials.

The principal hyperbolic domain PH D d consists of the space of all complex polynomials of degree d>0 whose filled Julia set is conjugate to a Jordan disc. In this paper, we characterize elements of PH D 3 and elements of its closure PH D 3 ¯ . Our results extend some works of Blokh, Oversteegen, Ptacek, and Timorin (The main cubioid, Nonlinearity. 27 (2014), pp. 1879–1897). On the other hand, we determine the basins of attraction for cubic polynomials, and we put technical conditions for a polynomial f to belong to PH D 3 . Next, we give some applications and some examples to illustrate our results.

Iterations of an analytic function of a complex variable is a wonderful laboratory for chaos. After classifying Mobius transformations by the nature of their fixed points, this chapter turns to iterations of rational functions. Mathematical notions include the Riemann sphere, metric spaces, and compactness. The Julia set of a map is the closure of its unstable periodic point. In essence, that is where the map is chaotic. The Fatou set is its complement. Some examples of Julia sets are plotted, as is the Mandelbrot set.

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x ( p , q ) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula and The following problem was posed by Paul Gauthier: if f (z) and g (z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f (z) and g (z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D . It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.