Approximation of pluricomplex Green functions: A probabilistic approach

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UPPSALA DISSERTATIONS IN MATHEMATICS

109

Approximation of pluricomplex Green functions

A probabilistic approach

Azza Alghamdi

Department of Mathematics

Uppsala University

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Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, 10134, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 21 September 2018 at 13:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Anna Zdunik (Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland.).

Abstract

Alghamdi, A. 2018. Approximation of pluricomplex Green functions. A probabilistic approach. Uppsala Dissertations in Mathematics 109. 47 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2714-5.

This PhD thesis focuses on probabilistic methods of approximation of pluricomplex Green functions and is based on four papers.

The thesis begins with a general introduction to the use of pluricomplex Green functions in multidimensional complex analysis and a review of their main properties. This is followed by short description of the main results obtained in the enclosed papers.

In Paper I, we study properties of the metric space of pluriregular sets, that is zero sets of continuous pluricomplex Green functions. The best understood non-trivial examples of such sets are composite Julia sets, obtained by iteration of finite families of polynomial mappings in several complex variables. We prove that the so-called chaos game is applicable in the case of such sets. We also visualize some composite Julia sets using escape time functions and Monte Carlo simulation.

In Paper II, we extend results in Paper I to the case of infinite compact families of proper polynomials mappings. With composition as the semigroup operation, we generate families of infinite iterated function systems with compact attractors. We show that such attractors can be approximated probabilistically in a manner of the classic chaos game.

In Paper III, we study numerical approximation and visualisation of pluricomplex Green functions based on the Monte-Carlo integration. Unlike alternative methods that rely on locating a sequence of carefully chosen finite sets of points satisfying some optimal conditions for approximation purposes, our approach is simpler and more direct by relying on generation of pseudorandom points. We examine numerically the errors of approximation for some simple geometric shapes for which the pluricomplex Green functions are known. If the pluricomplex Green functions are not known, the errors in Monte Carlo integration can be expressed with the aid of statistics in terms of confidence intervals.

Finally, in Paper IV, we study how perturbations of an orthonomalization procedure influence the resulting approximate Bergman functions. To this end we consider the concept of near orthonormality of a finite set of vectors in an inner product space, understood as closeness of the Gram matrix of those vectors to the identity matrix. We provide estimates for the errors resulting from using nearly orthogonal bases instead of orthogonal ones. The motivation for this work comes from Paper III: when Gram matrices are calculated via Monte Carlo integration, the outcomes of standard orthogonalisation algorithms are nearly orthonormal bases.

Keywords: pluricomplex Green function, pluriregular sets, Bernstein-Markov property,

Bergman function, nearly orthonormal polynomials, orthogonal polynomials, Monte Carlo simulation, composite Julia sets, Julia sets, iterated function systems, the chaos game.

Azza Alghamdi, Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden.

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List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Azza Alghamdi, Maciej Klimek

Probabilistic approximation of partly filled-in composite Julia sets. Annales Polonici Mathematici 119 (2017), 203-220 .

II Azza Alghamdi, Maciej Klimek, Marta Kosek

Attractors of compactly generated semigroups of regular polynomial mappings.

Complexity, (to appear in 2018). III Azza Alghamdi, Maciej Klimek

Approximation of pluricomplex Green functions based on Monte Carlo integration.

Manuscript.

IV Azza Alghamdi, Maciej Klimek, Marta Kosek

Bergman Functions and Nearly Orthonormal Polynomials. Manuscript.

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1. Introduction

The main mathematical object studied in this PhD thesis is the pluricomplex Green Function. It is the common theme for the four papers listed later in this thesis. More specifically, we investigate probabilistic methods of approx-imation of such functions. Our approach leads also to visualization of some pluricomplex Green functions and sets related to them.

1.1 Why to study the pluricomplex Green functions?

The pluricomplex Green functions are examples of particularly useful plurisub-harmonic functions. The concept of plurisubplurisub-harmonic functions was first in-troduced in 1942, independently by Kiyoshi Oka in Japan and Pierre Lelong in France (see e.g. [6]). Their discovery changed the multivariate complex analysis into one of the most dynamic branches of modern mathematical anal-ysis. There are two main reason for this. Since properties of plurisubharmonic functions are often similar to those of convex functions, many problems in pluricomplex analysis can be approached geometrically, or at least inspired by geometric properties of convex sets and functions. Also, plurisubharmonic functions are much easier to manipulate than the holomorphic functions which are the main object of interest in pluricomplex analysis. Over time, the branch of pluricomplex analysis focusing on properties of plurisubharmonic functions has become known as pluripotential theory.

The pluricomplex Green functions are examples of the so called extremal plurisubharmonic functions. In one complex variable they generalize classical Green functions with pole at infinity. In several variables, they are rather spe-cial among plurisubharmonic functions and provide a very powerful tool for investigation of properties of holomorphic functions, particularly those linked to polynomial approximation in several complex variables. The main reason for this is the fact that in many important situations these functions are just logarithms of Sicak’s polynomial extremal functions introduced in 1962 and indispensable in modern approximation theory (see [20]).

While pluricomplex Green functions belong to the world of pure mathe-matics, it should be mentioned that they have found an application in applied mathematics. It was shown by Ozan Öktem [16] that Siciak’s Separate An-alyticity Theorem (see Theorem C in Section 1.3), which uses pluricomplex Green functions in a significant way, can be used to characterize the range of the exponential Radon transform. This type of problem is fundamental for the techniques used in medical tomography.

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1.2 Preliminaries from pluricomplex analysis

If f : Ω −→ RN _{is a function from an open set Ω ⊂ R}M _{into R}N _{and a ∈ Ω,}

then f is said to be Fréchet differentiable at a, or simply differentiable at a if there is a linear map A : RM−→ RN_{such that}

lim

h→0

k f (a + h) − f (a) − A(h)k

khk = 0.

The mapping A is called the Fréchet differential (or Fréchet derivative) of f at a and will be denoted by daf. A specific choice of the norms in RM and

RN does not matter as all norms in a finite dimensional normed space are equivalent. If both M and N are even, say M = 2m and N = 2n, then f can be interpreted as a mapping from Ω ⊂ CMto CN, and then the Fréchet derivative daf can be decomposed into the C−linear part ∂af and the anti C−linear part

∂af, that is daf= ∂af+ ∂af, where daf = M

∑

j=1 ∂ f ∂ xj dxj+ ∂ f ∂ yj dyj = M

∑

j=1 ∂ f ∂ zj dzj | {z } ∂af + M

∑

j=1 ∂ f ∂ zj dzj | {z } ∂af ,

and the partial derivatives are evaluated at the point a. Here we used the stan-dard notations: dxj(z1, . . . , zM) = Re zj, dyj(z1, . . . , zM) = Im zj, dzj= dxj+ i dyj, dzj= dxj− i dyj, and ∂ ∂ zj =1 2 ∂ ∂ xj − i ∂ ∂ yj , ∂ ∂ zj =1 2 ∂ ∂ xj + i ∂ ∂ yj .

We say that f is holomorphic ( or analytic or C-differentiable) at the point a if daf exists and is a C−linear mapping ( that is, dafis additive and daf(λ z) =

λ daf(z), for all z ∈ CMand λ ∈ C). Equivalently daf= ∂af, which is the same

as to say that ∂af = 0.

If Ω ⊂ CM is open and non-empty, the mapping f : Ω −→ CN is said to be holomorphic ( or analytic or C- differentiable) in Ω, if it is holomorphic at each point in Ω.

The property of being holomorphic in an open set in several complex vari-ables is significantly different from the property of being differentiable in an open set in the sense of real analysis. There are many reasons for this but perhaps the main reason is expressed in the classic Hartogs’ Theorem on sep-arately analytic functions stating that if a function of several complex variables

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is separately holomorphic in an open set, then it is holomorphic in that set. Be-ing separately holomorphic (or separately analytic) in an open set means that the function is holomorphic with respect to each of the variables separately when the other variables are fixed.

We will denote the family of holomorphic functions on open set Ω1⊂ CM

with values in another open set Ω2⊂ CN byO(Ω1, Ω2), or simply byO(Ω1),

if Ω2 = C. If f ∈ O(Ω1, Ω2) is bijective and f−1∈O(Ω2, Ω1), we say that

f is biholomorphic. In the most general terms, the pluricomplex analysis can be described as the study of holomorphic functions as well as mathematical objects that are invariant with respect to biholomorphic mappings. To achieve this goal one uses many mathematical tools, among which the most important seem to be plurisubharmonic functions. Plurisubharmonic functions are the main object of study of the pluripotential theory. In this section, we just define such functions and state some of their basic properties. More information can be found in [8].

We need to recall a few definitions. We will use the usual symbols C, Ck, C∞_,

for continuous, continuously k-times differentiable and smooth functions, re-spectively. Also unless otherwise stated the symbols B(a, R) and B(a, R) will always denote the open and closed Euclidean balls, respectively, with centre at a and radius R.

Suppose that Ω ⊂ RN is open and u ∈ C2(Ω) is real valued. We say that u is harmonic in Ω if the Laplace equation is satisfied:

4u = N

∑

j=1 ∂2u ∂ x2_j ≡ 0 in Ω.

We will denote the class of harmonic functions on Ω by H(Ω).

Let Ω ⊂ RNbe open and non-empty. A function u : Ω −→ [−∞, ∞) is said to be upper semicontinuous if for each c ∈ R, the set {x ∈ Ω : u(x) < c} is open. A function u is said to be lower semicontinuous if −u is upper semicontinuous. Now suppose that u is an upper semicontinuous function such that u 6≡ −∞ on any connected component of Ω ⊂ RN. Then u is said to be subharmonic in Ω if for every relatively compact open subset G of Ω and every function h∈ C(G) ∩ H(G), the following statement is true:

u≤ h on ∂ G ⇒ u≤ h on G.

If u ∈ C2(Ω), then subharmonicity of u is equivalent to the inequality 4u ≥ 0 (see e.g. Theorem 2.5.1, [8]). We will denote the family of subharmonic functions on Ω by SH(Ω). Note that u ∈ H(Ω) if and only if u ∈ SH(Ω) and −u ∈ SH(Ω). It can be shown that subharmonic functions are locally absolutely integrable.

Now let Ω ⊂ CN be open and non-empty. An upper semicontinuous func-tion u : Ω −→ [−∞, ∞), such that u 6≡ −∞ on any connected component of Ω, is called plurisubharmonic in Ω, if for any a ∈ Ω and any b ∈ CN, the function

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λ 7−→ u(a + λ b) is subharmonic or ≡ −∞ on every component of the open set {λ ∈ C : a + λ b ∈ Ω}. The class of plurisubharmonic functions on an open set Ω is denoted by PSH(Ω). It can be shown that all plurisubharmonic functions are automatically subharmonic, so if Ω ⊂ CN, then PSH(Ω) ⊂ SH(Ω) and the inclusion is proper for N > 1.

It can be shown that, if u ∈ C2(Ω), then u is plurisubharmonic if and only if for each z ∈ Ω the complex Hesse matrix

∂2u ∂ zj∂ zk (z) of u is positive semidef-inite, that is, for all (a1, . . . , aN) ∈ CN we have

N

∑

j,k=1 ∂2u ∂ zj∂ zk (z)ajak≥ 0.

It can also be shown that, any subharmonic function can be approximated by a decreasing sequence of smooth subharmonic functions. The smoothing uses convolutions in the following way. If u : RN−→ [−∞, ∞] and v : RN_−→

[−∞, ∞] are two Borel functions, we define their convolution by the formula: (u ∗ v)(x) =

Z

RN

u(x − y)v(y)dλ (y),

for all values of x ∈ RN for which the integral exists, where λ denotes the Lebesgue measure.

Let h : R −→ R be defined by the formula h(t) =

exp(−1_t ) (t > 0)

0 (t ≤ 0).

Then h ∈ C∞_{(R). Now define}

ψ (x) = Ch(1 − kxk2) , x ∈ RM, where the constant C > 0 is chosen so thatR

RNψ (x)dλ (x) = 1, and we use the

Euclidean norm. Then, ψ ∈ C∞_(RM_{) and supp ψ is equal to the closed unit}

ball B(0, 1). We define then the standard smoothing kernels as the functions ψε(x) = 1 εMψ x ε , x_{∈ R}M, ε > 0. Clearly ψε ∈ C∞(R N_{) and moreover ψ}

ε(x) > 0 if and only if x ∈ B(0, ε). The

main approximation theorem for subharmonic functions ([8], Theorem 2.5.5) asserts that for any non-empty open set Ω ⊂ RN, any u ∈ SH(Ω) and any ε > 0 we have the following properties:

• u ∗ ψε∈ C∞∩ SH(Ωε), where Ωε = {x ∈ Ω : dist(x, ∂ Ω) > ε}, provided

that Ωε 6= /0.

• u ∗ ψε monotonically decreases with decreasing ε and

lim

ε →0

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Analogical statement is true if we replace RN_{by C}N_{and SH by PSH.}

Another general concept that we will need is that of a pluripolar set. We say that a set E ∈ CN is pluripolar, if for each point a ∈ E, there exists a neighbourhood U of a, and a function u ∈ PSH(U ) such that u(z) = −∞ for z∈ E ∩ U. Due to a classic result of Josefson (see e.g. [8]), E is pluripolar if and only if there exist a function u ∈ PSH(CN) such that

E_{⊂ {z ∈ C}N : u(z) = −∞}.

The Lebesgue measure of every pluripolar set E is zero, but these sets are also small in the sense that they are removable singularities for bounded from above plurisubharmonic functions. This is explained in the following theorem. Theorem: ([8],Theorem (2.9.22)) Let Ω be an open subset in CN, and let F be a closed pluripolar subset of Ω. If u ∈ PSH(Ω \ F) is bounded from above, then the functionu defined by the formula_e

e u(z) =

u(z) if z∈ Ω \ F,

lim supw→zu(w) if z∈ F,

is plurisubharmonic in Ω.

1.3 The pluricomplex Green function and pluriregularity

In a space of one complex variable (i.e., N = 1), one can define the classic Green function_{for a compact subset K ⊂ C with connected complement, as a} continuous function gK: C → [0, +∞) which is identically equal to zero on K

and harmonic on C \ K, with logarithmic pole at infinity, that is: gK(z) − log |z| = O(1) as z−→ ∞.

If such a function exists, the set K is said to be regular.

In the general case (i.e., N ≥ 1), letL (CN) denote the Lelong class, which consists of all plurisubharmonic functions u : CN −→ [−∞, ∞) with at most logarithmic growth at infinity, that is,

sup{u(z) − log(1 + kzk) : z ∈ CN} < ∞.

If E ⊂ CN is a non-empty set, we define the pluricomplex Green function (or the L− extremal function in older literature) of the set E as follows:

VE(z) = sup{u(z) : u ∈L (CN), u ≤ 0 on E}, z∈ CN.

A simple example of a pluricomplex Green function is the one for a closed ball B(a, R) with center a ∈ CN and radius R > 0, with respect to an arbitrary complex norm k . k. In this case

VB(z) = max 0, logkz − ak R .

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New examples can be produced from known ones with the help of Siciak’s product formula_{(see [23]): if E ⊂ C}Nand F_{⊂ C}M are compact sets, then

VE×F(z, w) = max{VE(z),VF(w)}, where z∈ CN, w ∈ CM.

Many more elaborate examples of pluricomplex Green functions will be given later.

The function VE is lower semicontinuous, if E is compact (see e.g.

Corol-lary 5.1.3 [8]). Let V_E∗denote the upper semicontinuous regularization of the pluricomplex Green function VE, that is

V_E∗(z) = lim

w→zsupVE(w), z∈ C N_.

The function VE∗is either inL (CN), if the set E is not pluripolar, or VE∗≡ ∞,

if E is pluripolar (see [23]).

Let E ⊂ CNbe a non-empty set and let a ∈ E. We say that E is pluriregular at the point a, if VE is continuous at a. A set E is called pluriregular, if it

is pluriregular at each point of its closure. In particular, if E is compact and V_E∗≡ 0 on E, then E is pluriregular (see e.g. Corollary 5.1.4 [8]).

In view of Siciak’s product formula, if both E and F are pluriregular, then so is E × F.

It is often non-trivial to decide if a set is pluriregular at a point or not. There are several criteria that can be used and one of the most useful among them is Ple´sniak’s analytic accessibility criterion [19]. We will state here a particu-larly convenient special case of that criterion as formulated by Pierzchała (see [18], Corollary 2.8):

If E _{⊂ C}N (or E_{⊂ R}N) is compact and a∈ E, suppose that there exists a polynomial γ : C −→ CN (or γ : R −→ RN) such that:

γ ((0, 1]) ⊂ Int E,

then E is pluriregular at γ(0), i.e., VE is continuous at γ(0).

In some applications of pluricomplex Green functions in approximation theory, one needs a stronger version of continuity. Let E ⊂ CN be a com-pact set. We say that E has the Hölder Continuity Property (or HCP) if there exist positive constants κ and µ, such that, if

dist (z, E) ≤ δ ≤ 1 then VE(z) ≤ κδµ.

It is a result of Zbigniew Błocki (presented in [25]) that HCP is equivalent to VE being Hölder continuous in the usual sense (and with the same constants),

that is

|VE(z) −VE(w)| ≤ κkz − wkµ, z, w ∈ CN.

LetP(CN) denote the space of all complex polynomials of N-complex vari-ables. The Siciak extremal function ([22] and [23]), is defined as

ΦE(z) = sup n |P(z)|deg P1 _{: P ∈}P(CN_{), kPk} E ≤ 1, deg P ≥ 1 o , z_{∈ C}N_,

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where k · kE denotes the supremum norm. The pluricomplex Green function

for any compact set E ⊂ CN can be expressed in terms of the Siciak extremal function (see [26] and [23]), namely we have the identity

VE≡ log ΦE. (1.1)

Recall that the polynomially convex hull bE _{of any compact set E ⊂ C}N is the set:

b

E= {z ∈ CN : |P(z)| ≤ ||P||E for all P ∈P(CN)}.

The set E is said to be polynomially convex if E = bE.

If N = 1, a compact set E ⊂ C is polynomially convex if and only if C \ E is connected. If E is a compact set in RN_{⊂ C}N_{, then E is polynomially convex}

because of the Stone-Weierstrass Theorem.

It follows directly from (1.1), that for any compact set E ⊂ CNwe have VE ≡ V_E_b.

Pluricomplex Green function is very useful in polynomial approximation the-ory, in particular, because of the fact that the continuity of ΦE (that is

plurireg-ularity of E) is equivalent to the Bernstein-Walsh -Siciak inequality (see [22]), which states that for each constant θ > 1, there exists an open set U containing E _{such that for each non constant complex polynomial P : C}N_{→ C we have}

|P(z)| ≤ kPkE θdeg P, z∈ U.

If E is pluriregular, then we can take U = {z ∈ CN: ΦE(z) < θ }.

In his ground-breaking paper [22] (see also [23]), Siciak applied this link to investigate a link between extensions of holomorphic functions and polyno-mial approximation. Here we present a reformulation of some of his results in the language of pluricomplex Green functions.

LetPn(CN) = {P ∈P(CN) : degP ≤ n}. Let E be a compact subset of CN.

For any continuous function f : E −→ C we define its uniform distance on E from polynomials of degree at most n by the formula (see [22], [23]):

ρn(E, f ) = inf{k f − PkE : P ∈Pn(CN)}.

We use the quantities ρn(E, f ) to define

ρ (E, f ) = lim sup

n→∞

ρn(E, f )1/n.

Let E ⊂ CNbe a pluriregular compact set. We put

DE,α =z ∈ CN : VE(z) < α , where α > 0. In [22] (see also [23]), Siciak proved the following three theorems:

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Theorem A: If E is pluriregular and g : DE,α −→ C is holomorphic for some

α > 0, then ρ (E, g) ≤ e−α.

Theorem B: If E is pluriregular, f : E −→ C is continuous and ρ(E, f ) < 1, then there exists a holomorphic function g: DE,α −→ C, where α =

− log ρ(E, f ), such that f ≡ g on E.

Theorem C: Let E ⊂ CNand F_{⊂ C}Mbe pluriregular and let α, β > 0. Let X = E × D_F,β ∪ (DE,α× F) .

Suppose that f_{: X −→ C is a separately analytic function on X, i.e.,} ∀z ∈ E, w7−→ f (z, w) is holomorphic in D_F,β

and

∀w ∈ F, z→ f (z, w) is holomorphic in DE,α.

Then there exists a unique holomorphic function ef in the open set Q= (z, w) ∈ DE,α× DF,β : VE(z) α + VF(w) β < 1 such that f = ef|X.

1.4 Explicit examples of pluricomplex Green functions

and pluriregular sets

The task of finding explicit formulas for pluricomplex Green functions or even showing that a particular compact set is pluriregular can be often difficult. In this section a few important examples will be presented. Details and references to the original papers can be found in [8].

First, we will define some needed notations. The symbol Extr(E∗) denotes the set of extreme points of the polar E∗of E, i.e., of the set

E∗= {y ∈ RN: hy, xi ≤ 1 ∀x ∈ E}.

For z = (z1, . . . , zN), w = (w1, . . . , wN) ∈ CN, we define the Hermitian inner

productas the product hz, wi = z1w1+ . . . + zNwN.

Now, we recall examples of the pluriregular sets and their pluricomplex Green functions:

(i) (Baran’s formula) If E ⊂ RN is an absolutely convex compact set, where 0 ∈ int(E), then

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where h is the inverse of the restriction to the set {z ∈ C : |z| > 1} of the Joukovski transformation: J(z) := 1 2 z+1 z , z_{∈ C \ {0}.}

Thus h : C \ [−1, 1] −→ {z ∈ C : |z| > 1} and it can be shown that h(t) = t +pt2_{− 1} _t_{> 1.} Consequently |h(z)| = h |z − 1| + |z + 1| 2 , z_{∈ C \ [−1, 1],} and lim z→w|h(z)| = 1, w∈ [−1, 1].

(ii) (Lundin’s formula) If B ⊂ RN _{denotes the closed unit Euclidean ball, then}

VB(z) =

1

2log h kzk

2_{+ |hz, ¯zi − 1| ,} _z

∈ CN. (1.3) If N = 1, the formula reduces to a well-known formula for V[−1,1]from

classic complex analysis.

(iii) Let P : CN _{−→ C}N _{be a complex polynomial mapping. The Łojasiewicz}

exponent at infinityL_∞(P) of the mapping P is the real number that is given by the formula:

L∞(P) = sup δ ∈ R : lim inf ||z||→∞ ||P(z)|| ||z||δ > 0 , (1.4)

for any fixed norm k . k. Interestingly, as shown in [21], any positive rational number is the Łojasiewicz exponent at infinity of a proper poly-nomial mapping1. Also, P is proper if and only ifL∞(P) > 0. A useful

method for producing examples is the following invariance result con-cerning polynomial mappings (see Theorem 5.3.1 [8]):

L∞(P)VP−1(E)≤ VE◦ P ≤ deg(P)VP−1(E) (1.5)

where E ⊂ CN and P : CN −→ CN _{is proper. In this case if E is}

pluri-regular, then so is P−1(E).

(iv) Let P : CN−→ CN_{be a proper complex polynomial mapping. If deg P = d}

and Pd is the homogeneous part of P of degree d, we say that P is a

regular polynomial mappingif inf {kPdk : kzk = 1} > 0. If P is regular,

thenL∞(P) = d and in particular

V_P−1_(E)= 1

dVE◦ P (1.6)

because of (1.5).

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(v) Also form (1.5), one can deduce an explicit formula for the pluricomplex Green functions of some analytic polyhedra [7]. Let P = (p1, . . . , pN) :

CN−→ CN be a complex polynomial mapping such that (p_b1, . . . ,pbN)

−1_{(0) = {0},}

wherepbkdenotes the homogeneous part of pk of maximal degree. Then the closed polynomial polyhedron

E_{= {z ∈ C}N : |pj(z)| ≤ 1, j = 1, . . . , N}

is pluriregular and it’s pluricomplex Green function is given by: VE(z) = max 0, 1 deg p1 log |p1(z)|, . . . , 1 deg pN |pN(z)| . (vi) In [2], Baran used (1.5) to calculate the pluricomplex Green function

for the standard simplex E = Conv (0, e1, . . . , eN) ⊂ RN ⊂ CN, where

{e1, . . . , eN} is the canonical basis in RN. Then z ∈ E if and only if

2z1− 1, . . . , 2zN− 1 and 2(z1+ . . . + zN) − 1 are in [-1,1] . Let P(z) =

(z2

1, . . . , z2N). Then P−1(0) = {0} and P−1(E) = B, where B is the closed

unit ball in RN. Then because of (1.5), the pluricomplex Green function of E satisfies the identity

VB(z) =

1 2VE(z

2

1, . . . , z2N),

and VB is known because of Lundin’s formula. Consequently, for

z= (z1, . . . , zN) ∈ CN we have VE(z) = log+h |z1| + . . . + |z1| + |z1+ . . . + zN− 1| .

1.5 The metric space of pluriregular sets

We will denote byR the family of all pluriregular polynomially convex com-pact subsets in CN. It was shown in [9] that this space is a complete metric space if it furnished with the metric:

Γ(E, F ) := max{kVEkF, kVFkE} for E, F ∈R.

We will refer to the metric space (R,Γ) as the space of pluriregular sets. Note that Γ(E, F) is also well defined if E, F are pluriregular compact sets, but are not necessarily polynomially convex. In this case, Γ is only a pseudometric.

It was shown in [9] that

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for any two pluriregular compact sets E, F ⊂ CN. In particular, Ej−→ E inR

if and only if VEj−→ VE uniformly in C

N_.

For a metric space (X, d), let δE(x) denote the distance from the point x ∈ X

to a subset E ∈ κ(X), where κ(X) denotes the set of all non-empty compact subsets of X. The classical Hausdorff metric χ on the set κ(X), is defined as follows:

χ (E, F ) := max{kδEkF, kδFkE}, E, F ⊂ X.

It can be shown that (κ(X), χ) is complete.

In particular, if X = CN with the Euclidean metric, then χ gives another metric onR. Unfortunately, despite a formal similarity between the definitions of Γ and χ, they are generating completely different topologies on the space of pluriregular sets (see [9] and [24]).

Let P : CN −→ CN _{be a proper polynomial mapping. If E ∈}_{R, then also}

P−1(E) ∈R and the function

A{P}:R 3 E 7−→ P−1(E) ∈R (1.7)

satisfies the Lipschitz condition with the constant 1/L∞(P) ( see Theorem 2

in [9]).

If R > 0, the symbol BR will be used to denote the closed Euclidean ball

with center at the origin and radius R. A radius of escape (or an escape radius) for a polynomial mapping P : CN−→ CN _{is a positive number R such that if}

z_{∈ C}N_{\ B}R, then limn→∞kznk = ∞, where z0= z and zn= P(zn−1) for n ≥ 1.

An escape radius may not exist in general, but ifL∞(P) > 1, an escape radius

can be found. Moreover, if δ ∈ (1,L∞(P)], then an escape radius R can be

chosen so that (see [10])

inf kP(z)k

kzkδ : kzk ≥ R

> R1−δ > 1. (1.8) We will call such an escape radius δ -adjusted. In this case

int (BR) ⊃ P−1(BR) = int (P−1(BR))

and

Γ P−1(BR), BR ≤

kPk_{∂ B}_R R δ .

In the rest of this section, we will assume thatF = {P1, . . . , Pk} is a finite

set of polynomial mappings Pj : CN −→ CN such that δ = min{L∞(P) :

P∈F} > 1 and that R > 1 is a δ-adjusted escape radius common to all Pj.

We can generalize the mapping A{P}from (1.7) in two different ways. We

define:

A_F(K) = [

P∈_F

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for any set K ∈R, where the hat denotes the operation of taking the polyno-mially convex hull of the set under it. According to [9] the mapping

HF:R 3 E 7→ HF(E) ∈R

is a contraction with contraction ratio 1/δ . And because of Banach’s fixed point theorem,H_F has a unique fixed point J[P1, . . . , Pk], which is called the

filled-in composite Julia set generated by P₁, . . . , Pk (see [9], [10]). In

particu-lar, ifF = {P}, then J[P] is the fixed point of A{P}. The set J[P] is sometimes

also called the autonomous filled-in Julia set of P, as it is directly related to the autonomous discrete dynamical system corresponding to iteration of the single polynomial mapping P : CN−→ CN_{. Moreover,}

J[P] =z ∈ CN: (Pn(z))∞n=1is bounded = lim_n→∞(P n

)−1(E), E∈R, where Pn= P ◦ . . . ◦ P (n times) and the limit is taken with respect to the metric Γ (see [9]).

The mappings A{P1}, . . . , A{Pk} form an iterated function system (IFS) on the metric space (R,Γ). We will denote to the attractor of this IFS by S = S[P1, . . . , Pk]. This set is the unique fixed point of the mapping

κ (R) 3 K 7−→

k

[

j=1

A{Pj}(K) ∈ κ(R).

So the set S is a compact subset of R, and its structure is important for un-derstanding the relationship between the various Julia sets that arise in this context.

To explain this, we need to use iterations which can apply different maps at each step of iteration. For this, a labeling system is useful. If k ≥ 2 is an integer, the space of full addresses is the set Σk of all functions σ : N −→

{1, . . . , k} equipped with the metric d(σ , τ) = ∞

∑

j=1 |σ ( j) − τ( j)| kj , σ , τ ∈ Σk. (1.9)

Any function σ : {1, . . . , m} −→ {1, . . . , k} will be called a partial address of length m. If σ ∈ Σkand m ∈ N, we associate with the full address σ , the partial

address of length m given by:

σ |m = σ

{1,...,m}.

It is easy to characterize the metric d: • if d(σ , τ) < k−m_{, then σ |m = τ|m;}

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The metric space (Σk, d) is compact, with a base for its topology given by sets

of all full addresses with a common fixed partial address.

For any z ∈ CN and σ ∈ Σk we define a σ -orbit of z (with respect to

P1, . . . , Pk) as the infinite sequence z0, z1, z2, . . ., where z0= z, and

zn= Pσ (n)(zn−1), n∈ N.

In the case of a partial address σ of length m, we define in a similar way a truncated σ -orbit of the point z (of length m + 1).

Define Jtr[P1, . . . , Pk] as the set of all z ∈ CN such that for each m ∈ N there

exists σ ∈ Σksuch that the (σ |m)-orbit of z is contained in BR. It can be shown

that the set Jtr[P1, . . . , Pk] is compact and pluriregular2 (see [12] and [13]). It

turns out that J[P1, . . . , Pk] is the polynomially convex hull of Jtr[P1, . . . , Pk] (see

[12]). We will call the set Jtr[P1, . . . , Pk] the partly filled-in composite Julia set

generated by P1, . . . , Pk.

If σ ∈ Σk and E ∈R, then the following set is an element of R which is

independent of the choice of E (see [10]): Sσ = lim

m→∞(Pσ (m)◦ . . . ◦ Pσ (1)) −1_(E),

where the limit is taken with respect to the metric Γ. The set Sσ can be

de-scribed as the set of all points z ∈ CN whose σ -orbits are bounded, and hence it make sense to call it the non-autonomous filled-in Julia set corresponding to σ and P1, . . . , Pk, as it can be linked to the non-autonomous discrete dynamical

system CN, {P_{σ (m)}}m∈N.

It turns out that the set

S = S[P1, . . . , Pk] = {Sσ : σ ∈ Σk} ⊂R

(see [10]), and hence by Theorem 1(a) in [1] the set Jtr[P1, . . . , Pk] = [ σ ∈Σk Sσ = [ S

is compact in CN (see also [13]). The unionSS does not have to be

poly-nomially convex (see [10] or [15]).

For any m ∈ N, we will define an m-outline as any set of the form S[σ ] = (Pσ (m)◦ . . . ◦ Pσ (1))

−1

(BR),

where σ : {1, . . . , m} −→ {1, . . . , k} is any partial address. Given σ ∈ Σk, we

have: S_σ = lim m→∞S[σ |m] = \ m≥m₀ S[σ |m], m0∈ N, (1.10)

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where the limit can be taken either with respect to the metric Γ inR (see [10]) or with respect to the ordinary Hausdorff distance between compact sets in CN associated with the Euclidean metric.

We will denote the set of all m-outlines bySm. Obviously

Sm= {S[σ |m] : σ ∈ Σk} ⊂R,

and the number of elements of the set Sm is at most km. Also, we have the

convergence lim m→∞ [ Sm= [ S

with respect to the ordinary Hausdorff distance on the set κ(CN). For the pur-pose of visualization it is useful to name the setSS

m∈ κ CN the cumulative

m-outlineof the partly filled-in composite Julia setSS.

1.6 The complex Monge-Ampére Operator

In one complex variable, harmonic functions can be described as maximal subharmonic functions in the following sense. Suppose that u ∈ SH(Ω), where Ω ⊂ C is open. Then u is harmonic in Ω (i.e. ∆u ≡ 0) if and only if for any relatively compact domain ω ⊂ Ω and any v ∈ SH(ω) ∩ C( ¯ω ), if v ≤ u on ∂ ω , then v ≤ u in ω. Note that if u ∈ SH(Ω), then ∆u can be calculated in the sense of distributions ( see e.g. [8]).

Let Ω ⊂ CN be a nonempty open set and let u ∈ PSH(Ω). By analogy with the one dimensional situation, we can say u is a maximal plurisubharmonic functionif for any relatively compact domain ω ⊂ Ω and any v ∈ PSH(ω) ∩ C( ¯ω ), if v ≤ u on ∂ ω , then v ≤ u in ω . If N > 1, maximality can also be characterized by a differential operator but the situation is more complicated, because in contrast to the Laplacian this operator is non-linear.

For an open set Ω ∈ CN and u ∈ PSH ∩ C2(Ω), it can be shown that u is maximal in Ω if and only if

det ∂2u ∂ zj∂ zk ≡ 0 in Ω.

If u is pluriharmonic function ( i.e., u ∈ C2(Ω) and ∂

2_u

∂ zj∂ zk

≡ 0, where j, k = 1, . . . , N), then obviously u satisfies this criterion, but there exist many maximal plurisubharmonic functions which are not differentiable. Note also that in one complex variable

∆ = 1 4

∂2u ∂ z∂ z.

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The above characterization of maximality can be extended to non-differentiable functions as follows. First if that u ∈ PSH ∩ C2(Ω), we define the non-negative measure M[u] = 4NN! det ∂2u ∂ zj∂ zk dλ ,

where λ = λ2N is the 2N− dimensional Lebesgue measure in CN= R2N. Bedford and Taylor proved in 1982 [3], that the definition of M can be ex-tended to the case of locally bounded plurisubharmonic functions by adopting the following definition.

Let u : Ω −→ R be a locally bounded plurisubharmonic function. If V ⊂ Ω is open, we define M[u](V ) = sup ϕ lim ε →0 Z Ω ϕ dM[u ∗ ψε] ,

where the supremum is taken over all continuous functions ϕ : V −→ [0, 1] with compact support and ψε denotes a standard smoothing kernel. If E is a

Borel subset of Ω we put M[u](E) = inf

n

M[u](V ) : V is open , E ⊂ V ⊂ Ω o

.

The operator u 7−→ M[u] is called the (generalized) complex Monge-Ampére operator. If M[u](Ω) = 0, we say that u satisfies the (complex) homogeneous Monge-Ampére equationin Ω.

The key property of the operator M, which was shown in [3], states that a locally bounded plurisubharmonic function u in Ω is maximal if and only if u satisfies the homogeneous Monge-Ampére equation in Ω.

If E ⊂ CN is bounded and non-pluripolar, then the measure M[V_E∗] is called the (complex) equilibrium measure for E. It can be shown, that M[VE∗](CN\

¯

E) = 0 and M[V_E∗]( ¯E) = (2π)N. Moreover, the pluricomplex Green function is the unique function u ∈ PSH(CN), which vanishes on E, except possibly on a pluripolar set, satisfies the homogeneous Monge-Ampére equation in CN\ ¯E and is such that

u(z) − log kzk = O(1) as kzk → ∞.

One very useful application of the Monge-Ampére operator is to lead to a way of detecting on pluripolar sets.

1.7 Two capacities

Recall that if Ω is an open subset in CN and 2Ω_{denotes the set of all subsets}

of Ω, then a function Cap : 2Ω _{→ [0, ∞) is called a generalized capacity if it}

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• Cap( /0) = 0;

• if E1⊂ E2⊂ . . . ⊂ El ⊂ . . . ⊂ Ω, then supj∈NCap(Ej) = Cap(SjEj);

• if K1⊃ K2⊃ . . . ⊃ Kl. . . is a sequence of compact subsets of Ω, then

Cap(\

j

Kj) = lim

j→∞Cap(Kj).

Recall also that a bounded domain Ω ⊂ CN is said to be hyperconvex if there exists a continuous plurisubharmonic function ρ : Ω −→ (−∞, 0) such that for every number c < 0 the set {z ∈ Ω : ρ(z) < c} is relatively compact in Ω. For example, if Ω = B(a, R) is the open Euclidean ball with center at a_{∈ C}N and radius R > 0, then we can take ρ(z) = log(kz − ak/R), to verify that Ω is hyperconvex.

Assume that Ω is hyperconvex. For any set E ⊂ Ω we define the (outer) relative capacityof E (with respect to Ω) by the formula

C(E, Ω) = inf

ω

n

sup {M[u](ω) : u ∈ PSH(Ω), 0 < u < 1} o

where the infimum is taken over all open sets ω such that E ⊂ ω ⊂ Ω. If E is compact we simply have

C(E, Ω) = sup n

M[u](E) : u ∈ PSH(Ω), 0 < u < 1 o

.

It was shown in [3] that this is a generalized capacity and what is more, C(E, Ω) = 0 if and only if E is pluripolar. Yet another property is subadditivity: if E1, E2, E3, . . . ⊂ Ω, then C ∞ [ j=1 Ej, Ω ! ≤ ∞

∑

j=1 C(Ej, Ω) .

If in a particular context the set Ω is fixed, it is customary to write C(E) instead of C(E, Ω).

Many different capacities can be defined in CN, but perhaps the best known is the logarithmic capacity generalizing a similar concept from the complex plane (see [23]). Let E be a subset of CN, we define the Robin constant of E as

γ (E) = lim sup

kzk→∞

(VE∗(z) − log kzk) .

Then the logarithmic capacity of E is given by c(E) = exp(−γ(E)). Because of the general properties of the pluricomplex Green functions, c(E) = 0 if and only if E is pluripolar. It can also be shown that the logarithmic capacity is actually a generalized capacity [14].

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1.8 Bernstein-Markov property and Bergman functions

Bernstein-Markov property is a tool to compare asymptotically the L∞_-norms

(uniform norms) and L2-norms on a given set E of polynomials of a given degree, as the degree is allowed to go to infinity. The reason it is used in this dissertation is the fact that without it the kind of approximation of the pluricomplex Green functions that we want to use would not work.

Let E ⊂ CN be a pluriregular compact set. The pair (E, µ), where µ is a positive finite measure such that supp µ = E, has a Bernstein-Markov prop-erty, if the numerical sequence

Mk= sup ( ||p||E ||p||_L2_(µ) : p ∈Pk(CN) ) , k≥ 1, (1.11)

has subexponential growth, that is lim sup

k→∞

k p

Mk≤ 1. (1.12)

For example, if E is a pluriregular compact set, then the pair (E, M[V_E∗]) satisfies the Bernstein-Markov property (see [5]).

A sufficient condition for the Bernstein-Markov property is the mass-density criterionthat was proposed by Bloom and Levenberg in [4]. Let E be a pluri-regular compact subset of the open unit ball B(0, 1) in CN. Let C(·) denote the relative capacity function in the unit ball in CN (so C(F) = C (F, B(0, 1)), for F⊂ B(0, 1), using the notation from the previous section). If for some positive constant T the condition

lim

r→0+C {z : µ ( ¯B(z, r)) ≥ r

T_{} = C(E)} _(1.13)

is fulfilled, then the pair (E, µ) has the Bernstein-Markov property.

The Bernstein-Markov property needed in approximation of the pluricom-plex Green functions by so called Bergman functions, which are simple func-tions constructed from finitely many polynomials. They can be described as follows.

Let E ⊂ CNbe a compact set. We will be assuming that E is determining for complex polynomials, that is, the only polynomial which is identically zero on E is the zero polynomial. Let Nk= dimPk(CN). For a positive Borel measure

µ supported on the set E ⊂ CN, consider an orthonormal basis {Qj∈Pk(CN) :

j= 1, . . . , Nk} for the spacePk(CN) ∩ L2(µ) with respect to the inner product

given by:

hP, Qi =

Z

E

P(z)Q(z) dµ(z), P, Q ∈Pk(CN).

We define the Bergman function Bµ_k of order k (or simply the k-th Bergman function) of the set E by the formula:

Bµ_k(z) =

N_k

∑

j=1

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Using Parseval’s identity, it can be shown that this definition is independent of a specific choice of the orthonormal basis. Note that the functions _2k1 log Bµ_k are in the Lelong classL (CN) for all k.

It was shown by Bloom and Shiffman in [5] that if E is pluriregular and the pair (E, µ) has the Bernstein-Markov property, then the pluricomplex Green function of E can be approximated by the formula

lim k→∞ 1 2klog B µ k(z) = VE(z), z∈ C N . (1.14)

The convergence is uniform on compact subsets of CN ( Lemma 3.4 [5] or Theorem 1.2.1 in [17]). We will refer to (1.14) as the Bloom-Shiffman ap-proximation formula. The formula shows that in order to approximate the pluricomplex Green function of a compact set it suffices to devise means of approximating the Bergman functions of this set.

1.9 Monte-Carlo integration

Since two of the papers forming this thesis use Monte-Carlo methods, this section will provide a very brief introduction to this subject.

Let µ be a probabilistic Borel measure supported on a closed set F ⊂ RM. If X is a random variable with values in F whose probability distribution is µ, then for any Borel function g : F −→ R we have

E[g(X )] =

Z

F

g(x) dµ(x),

provided that the expected value (or equivalently the integral on the right) exists. Suppose that X1, X2, . . . are independent samples from µ (i.e.

indepen-dent replications of X ). The sample mean and the sample standard deviation of g(X ) are then given respectively by

mn= 1 n n

∑

j=1 g(Xj) and sn= s 1 n− 1 n

∑

j=1 (g(Xj) − mn)2

for n = 1, 2, 3, . . . . By the Strong Law of Large Numbers they converge, re-spectively, to E[g(X )] andpVar[g(X)] (with probability 1 and in L1). By the Central Limit Theorem

mn− E[g(X)]

sn/

√

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in distribution, where N(0, 1) denotes the standard Gaussian distribution. If Φ is the cumulative distribution function for N(0, 1), that is

Φ(x) =√1 2π

Z x

−∞

e−y2/2dy,

then for θ ∈ (0, 1) the number zθ = Φ−1(θ ) is called the θ -quantile of the

standard normal distribution. If we choose a significance level θ ∈ (0, 1), then the corresponding confidence interval for E[g(X )] with confidence level (1 − θ )100% is mn− z1−θ /2 sn √ n, mn+ z1−θ /2 sn √ n .

The absolute width of this confidence interval is its length (which in this case is random) and the relative width is the ratio of the absolute width to mn. The

confidence level expresses the proportion of confidence intervals containing the true value of E[g(X )] if the sampling experiment was repeated indepen-dently infinitely many times.

Our numerical experiments using Monte Carlo simulations are conducted with two goals in mind. One goal is to calculate the Gram matrices to ap-proximate orthonormal bases for the considered spaces, and thus to approxi-mate the Bergman functions and hence the corresponding pluricomplex Green functions. In this context Monte-Carlo simulations provide a way of calcu-lating large number of integrals. The other goal, is to draw images for some composite Julia sets, with the help of the escape time functions. In the case of composite Julia sets, if one wants to perform this task deterministically, the number of orbits to consider is so large that an average computer would need a prohibitively long time to complete the calculations. The Monte-Carlo approach reduces the number of calculations considerably, making the task computationally viable.

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2. Results of the Thesis

2.1 On Paper I

As stated in Section (1.5), the topology of the metric space (R,Γ) of all com-pact, polynomially convex and pluriregular subsets of CN is complicated and not well understood. In this paper we first prove some general properties of this space (Theorem 1). Then we focus on probabilistic approximation of the composite Julia sets, which form a dense subset of the spaceR (Theorem 2), and show how composite Julia sets can be efficiently visualised with the help of Monte Carlo simulation (Theorem 3).

IfF is a family of sets, we will use the notation

[

F = [

F∈_F

F. Otherwise we use the notation from earlier sections.

We prove the following properties for the space (R,Γ). Theorem 1. The space (R,Γ) has the following properties:

(a) IfK ⊂ R is compact, then SK is compact in CN_.

(b) IfA,B ⊂ R are non-empty and compact, then Γ

_[ A,[

B≤ χΓ(A,B).

(c) For every m ∈ N, the mapping Rm 3 (C1, . . . ,Cm) 7→ \m [ j=1 Cj∈R

is continuous, where the hat denotes the operation of taking the polyno-mially convex hull.

(d) The space (R,Γ) is separable, but is not proper, i.e. closed balls do not have to be compact.

The fact that R is not proper is interesting, as it shows that some earlier results concerning iterated function systems in metric spaces do not apply in our case.

In the following theorem, we show that Jtr[P1, . . . , Pk], the partly filled-in

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in a probabilistic manner by a sequence of randomly chosen iterates of any chosen element ofR.

Theorem 2. Consider polynomial mappings P1, . . . , Pk : CN −→ CN with

Łojasiewicz exponents greater than 1.

(a) IfC ⊂ R is bounded and a sequence of partial addresses τm: {1, . . . , m} → {1, . . . , k},

with m∈ N, is given, then any dilation of S in R contains almost all of the sets Cm= P_τm₍₁₎◦ . . . ◦ P_τm_(m) −1 (C) : C ∈C , m_{∈ N.} (b) LetU ⊂ R be an open set such that U∩S 6= /0 and let n ∈ N. There exists

a partial address θ of length m ≥ n and ε > 0 such that the image of the ε -dilation ofS via the mapping

R 3 F 7−→ Pθ (1)◦ . . . ◦ Pθ (m)

−1

(F) ∈R is contained inU.

(c) Let an address τ ∈ Σk be generated according to a set of

probabili-ties p₁, . . . , pk> 0 such that p1+ . . . + pk= 1, that is, the values τ( j)

of τ are selected at random, independently from each other, so that Pτ ( j) = i = pi, for j∈ N and i ∈ {1,. . . , k}. Then for any E ∈ R,

lim m→∞Γ Jtr[P1, . . . , Pk], [ Em = 0 (2.1)

with probability one, where Em=

n

P_{τ (1)}◦ . . . ◦ P_{τ (n)}−1

(E) : n ≥ mo (2.2) and the closure ofEmis taken inR.

Due to the complicated geometry of the Γ−convergence, the theorem can not be applied directly to generate an image of a partly filled-in composite Julia set using the computer. To this end, we need an approximation with respect to the Hausdorff distance associated with the underlying Euclidean metric.

If σ ∈ Σk, then the m-outline is defined as the set

S[σ |m] = (Pσ (m)◦ . . . ◦ Pσ (1)) −1

(BR).

We will denote the set of all m-outlines bySm. Note that

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Let’s define the escape time function for the cumulative m-outline of our Julia set by the formula

tm(z) = m

∑

i=1 1S Si(z), where m ∈ N, z ∈ BR.

For any j ∈ {1, . . . , m} this function’s j-th superlevel set {z ∈ BR:tm(z) ≥ j}

is exactly the cumulative j-outline SS

j of the Julia set Jtr[P1, . . . , Pk]. For

approximation purposes, we consider a version of the escape time function which does not take into account all the orbits of points in BRof length m, but

only specific selection of such orbits. Let

/0 6= Λ ⊂ {1, . . . , k}{1,...,m}. We define for z ∈ BR tΛ m(z) =    0, if ||Pλ (1)(z)|| > R for all λ ∈ Λ,

max j ≤ m : ||(Pλ ( j)◦ . . . ◦ Pλ (1))(z)|| ≤ R, for some λ ∈ Λ ,

otherwise. We can describetΛ

mas being a partial escape time function. ClearlytΛm≤tm

for any choice of Λ, and if Λ = {1, . . . , k}{1,...,m}, thentΛ m≡tm.

Using Monte- Carlo methods, we can visualize Julia sets, by calculating the escape time functions for the cumulative m-outline of our Julia set. In fact, the setsSS

j form a decreasing sequence that converges in both with respect to Γ

and with respect to the ordinary Hausdorff metric in a space of the compact sets in CN to that Julia set.

Theorem 3. Assume that P1, . . . , Pk: CN −→ CN are polynomial mappings

with Łojasiewicz exponents greater than 1, and that R is a common escape ra-dius for these polynomials, adjusted to the minimum of their exponents. Given m_{∈ N, the escape time function t}mfor the cumulative m-outline of the partly

filled-in composite Julia set Jtr[P1, . . . , Pk] can be approximated as follows. Let

D_{⊂ C}N be a domain and let µ be a Borel probability measure whose support is ¯D. Let ν be a strictly positive probability mass function on the set of par-tial addresses{1, . . . , k}{1,...,m}_{. Fix L}_{, M ∈ N. Consider independent random}

variables Z ∼ µ and λ1_{, . . . , λ}L_{∼ ν (with probability distributions µ and ν,}

respectively). Then producing a sequence of samples n

(zi,ti) ∈ ¯D× {0, . . . , m} : i = 1, . . . , M

o

(2.3) of the random variable

Z,tΛ

m(Z)

, where Λ = {λ1, . . . , λL}and letting M→ ∞, we approximate the graph of the escape time functiontmin the

fol-lowing sense. For any j∈ {1, . . . , m}, the discrete superlevel sets {zi∈ ¯D: ti≥ j, i = 1, . . . , M}

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converge in the usual Hausdorff metric inκ(CN), with probability one, to the corresponding superlevel sets_{{z ∈ ¯D :} m(z)≥ j} of the escape time functions,

which coincide with the matching cumulative j-outlines of the considered Julia set.

The last theorem can be used to visualize composite Julia sets. Figures 2.1, 2.2 and 2.3 show some examples. In the case of the Figures 2.1 and 2.2 we have a deterministic plot on the left and the escape time function contour plot on the right. Both sets are polynomially convex. Figure 2.3 shows two visualizations of polynomially non-convex composite Julia sets, which were mathematically described in earlier literature. This time the plots on the left are Monte Carlo generated, and the plots on the right are again the contour plots of the escape time functions of the respective sets.

Figure 2.1. A composite Julia set – the attractors of the two generating polynomials are the line segment[_{−2,2] and the unit disc, respectively.}

Figure 2.2. A composite Julia set – the attractors of the two generating polynomials are the line segments[_{−2,2] and [−2i,2i], respectively.}

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2.2 On Paper II

In this paper we prove a counterpart of Theorem 2 from Paper I, but for com-posite Julia sets generated by infinite compact families of regular polynomial mappings of a fixed degree. In this case, some topological aspects get more complicated than in the case of finite families of polynomials.

Let us adopt a few notational conventions. Let N ∈ N be fixed and let P denote the vector space of all polynomial mappings P : CN _{−→ C}N_{. Let}

Pd= {P ∈P : degP = d} and for d ≥ 2 let P∗d= {P ∈Pd : P is regular}.

The paper begins with a series of general observations concerning the space (R,Γ) as we listed next.

(1) IfG ⊂ R is compact, then

[

G =[

G,

but for non-compact setsG this is not necessarily true.

(2) If P ∈P∗_d, then the mapping AP:R 3 K 7→ P−1(K) ∈R is a contractive

similitude with the contraction ratio1/d. (3) The mapping

P∗

d×R 3 (P,K) 7→ P−1(K) ∈R

is continuous with respect to the product topology onP∗_d×R. (4) LetF be a non-empty compact subset of P∗_d. The mapping

AF: κ(R) 3 K 7→ [ P∈_F

AP(K) ∈ κ(R)

is well defined and is a contraction with ratio1/d. In particular, the mappingA_Fhas a unique fixed pointS[F] ∈ κ(R).

IfF is a non-empty compact subset of P∗_d, then there exist an escape radius R common to all mappings in F. In particular, this leads to the following definition. For any sequence (Pn)∞_n=1⊂F we define its filled-in Julia set

(non-autonomousif the sequence is not constant) as:

J[(Pn)∞n=1] =z ∈ CN: ((Pn◦ ... ◦ P1)(z))∞n=1 is bounded .

It is shown that, for any E ∈R we have the convergence J[(Pn)∞n=1] = lim_n→∞(Pn◦ ... ◦ P1)−1(E) = lim

n→∞J[Pn◦ ... ◦ P1]. Then J[(Pn)∞n=1] = \ n≥1 (Pn◦ ... ◦ P1)−1(BR).

We define the partly filled-in composite Julia set of the compact familyF as Jtr[F] = \ m∈N " [ P₁,...,Pm∈F (Pm◦ ... ◦ P1)−1(BR) # .

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This set is compact and its polynomially convex hullJ[F] is the unique fixed point of the mapping

R_{K →}

P∈F

AP(K)∈ R.

The setJ[F] is called the filled-in composite Julia set of F.

The following theorem describes the connection between the non-autonomous Julia sets and the attractor S[F].

Theorem 1. Let F be a non-empty compact subset of P∗_d. Then (1) S[F] =J[(Pn)∞_n=1] : (Pn)∞_n=1∈ FN

; (2) Jtr[F] =S[F].

To illustrate the additional level of complexity when dealing with Julia sets generated by infinite families of polynomial mappings, one could look at the following one-dimensional case. For c∈ C, let pc(z) = z2+ c. Let c0= 0.3 +

0.5i and L =_{c0+ a + ib : a, b∈ [−0.1,0.1]}. Let F = {pc : c∈ L}. Figures

2.4 and 2.5 show a selection of sets J[pc] for pc ∈ F. Figure 2.6 shows an

approximate outline of the setJtr[F]. The gray scale picture reflects better the

probability distribution of the points in this set.

Figure 2.4.Plots of the autonomous filled-in Julia sets J[pc], with c = c0and 11 values

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Figure 2.5. The set J[pc0] and 11 non-autonomous Julia sets J[{pcj}∞j=1], with

se-quences_{cj} selected at random from L.

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The next theorems are counterparts of Theorem 2 from Paper I, in the case of infinite compact families of regular polynomial mappings.

Theorem 2. LetF ⊂ P∗_dbe non-empty and compact.

(a) If (πn)∞_n=1⊂F, E ∈ R and U ⊃ S[F] is an open subset of R, then almost

all elements of the sequence

E = {(Aπn◦ . . . ◦ Aπ1) (E) : n ≥ 1} ,

belong to U. In particular, all accumulation points of this sequence belong toS[F] and thus E is compact in R.

(b) If E ∈S[F] and V ⊂ R is a neighbourhood of E, then there is an open set U ⊃ S[F] and there exist mappings Q1, . . . , Qm∈F, such that

(AQm◦ . . . ◦ AQ1) (U) ⊂ V. The value of m can be made arbitrarily large.

Theorem 3. LetF be a non-empty compact subset of P∗_dand letF0= {πn: n ∈

N} be a dense countable subset of F. Let τ : N → N be generated according to probabilities p₁, p2. . . > 0 such that ∑∞n=1pn= 1, i.e. the values τ( j) of τ

are chosen at random, independently from each other, so that P [τ( j) = i] = pi

for i_{, j ∈ N. Let} Em= n πτ (1)◦ . . . ◦ πτ (n) −1 (E) : n ≥ m o , m_{∈ N.} Then for any E∈R, with probability one

lim m→∞Γ _[ S[F],[ Em = 0. Also, a deterministic version of this theorem is shown.

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2.3 On Paper III

In this paper, we approximate the pluricomplex Green functions VE of

pluri-regular compact sets E ⊂ CN that satisfy the Bernstein-Markov property (see formulas (1.11) and (1.12)) with a positive measure µ, using Monte Carlo methods. The Monte Carlo integration is used as means of numerical imple-mentation of the Bloom-Shiffman formula (1.14) (see Section 1.8).

The Bloom-Shiffman formula shows that the Bergman function of suffi-ciently high order can provide a good approximation of the corresponding pluricomplex Green function. However, in some cases one has to choose a very high order to get a reasonable quality approximation of their Green func-tions. As an example of this, we calculate the Bergman functions for the n-asterisk set defined as the compact set An= {z ∈ C : zn∈ [−1, 1]}, where

n≥ 2, ( see Figure 2.7).

-1 0 1

Figure 2.7.The n-asterisk with n = 11.

The set An is (pluri)regular and a formula for the Green function VAn can be calculated from the invariance formula (1.6) and the known formula for V_[−1,1](see Section 1.4). The natural probability measure µ on Anis based on

normalized arc length:

µ (S) = 1 2n n

∑

m=1 Z [−am,am] 1S(z) |dz|,

for any Borel subset S ⊂ An. The mass-density criterion in Section 1.8, implies

that the pair (An, µ) has the Bernstein-Markov property.

The Bergman functions of order k < 2n can be calculated explicitly:

B_kµ(z) =

k

∑

j=0

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Figure 2.8. The graph of the log-normalized Bergman functions of order k= 22 for A11(left) and the Green function VA11(right).

Clearly these functions are constant on any circle centered at the origin and hence cannot provide a good approximation of VAn. Figure (2.8) shows that even at k= 2n there is no discernible progress in approximation, whereas for k= 40 we get a graph that looks practically like the exact Green function.

The range of sets for which we calculate approximate Green functions in this paper are all closures of strongly Lipschitz domains in RN ⊂ CN_{, and}

as such they are pluriregular, polynomially convex and have the Bernstein-Markov property. We use spherical segments within the considered set, with apexes on the boundary, to show that this is the case.

Our approach for approximation of the pluricomplex Green functions re-lies on the fact that, with probability one, the Bergman functions of order k can be approximated by replacing the measure µ by a discrete measure sup-ported on a finite subset of E consisting of points in E drawn at random from the probability distribution given by µ. We state this result in the following theorem.

Theorem: Letµ be a probability measure supported on a determining com-pact set E_{⊂ C}N. For a sequence of pointsξ1,ξ2, . . . in E drawn independently

at random according to the probability distributionµ, define the discrete prob-ability measures µn= 1 n n

∑

j=1 δ_ξ_j, n_{∈ N.} (2.4)

Then with probability one

Bµ_k = lim

n_→∞B

µn

k , k∈ N, (2.5)

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In computational terms, the basic idea here, can be outlined as follows. Given a polynomial basis q1, . . . , qN_k ofPk(Cn), the Gram matrix

Gram(q1, . . . , qNk) = h hqi, qjiµ i i, j=1,...,Nk ∈ CNk×Nk_,

is needed as input of an orthonormalization algorithm. The resulting orthonor-mal basis for the space Pk(Cn) ∩ L2(E, µ) is then used to form the Bergman

function of order k. However, instead of calculating exactly the inner products in the Gram matrix, we rely on Monte-Carlo integration to get approximate values. And then the orthonormalization algorithm utilizes this approximate Gram matrix.

The Strong Law of Large Number and the Central Limit Theorem ensure that the more pseudorandom points we are generating for Monte-Carlo inte-gration, the better approximation of the pluricomplex Green function we get.

In most examples considered in this paper, the exact formulas for the inves-tigated pluricomplex Green functions are known, and so we can examine the errors of the Monte-Carlo approximation, either through visualization or by tabulating the Root Mean Square Error (RMSE) in a domain of choice.

The paper addresses also a few implementation issues, like finding appro-priate densities for importance sampling, suitable parametrization of portions of the considered sets facilitating generation of pseudorandom points, and identification of extreme points of the duals of convex sets.

In the following Figure (2.9), we show visualization of the exact pluri-complex Green functions obtained by known formulas such as Lundin’s and Baran’s formulas for the Green function for sets in C2. In Figure (2.10), we visualize the approximate pluricomplex Green functions for the same sets by using the Bloom-Shiffman approximation formula (1.14). In figures (2.11) and (2.12), we show plots of the approximate Green function for sets for which explicit formulas of their pluricomplex Green functions are not directly avail-able. To be precise, we show the approximate Green function VLof the lens-set

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Figure 2.9. Plots of the explicit pluricomplex Green functions for the unit disc, the square[_−1,1]2_{, the square with vertices}_{(1, 0), (0, 1), (}

−1,0),(0,−1) and the regular

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Figure 2.10. Plots of the approximate pluricomplex Green functions for the unit disc, the square[_−1,1]2_{, the square with vertices}_{(1, 0), (0, 1), (}

−1,0),(0,−1) and the

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Figure 2.11. The lens-set L= D ((−1,0),3/2) ∩ D((1,0),3/2) ⊂ R2_{(top), a}

Monte-Carlo approximation of the VL (bottom left) and the exact VL (bottom right), both

restricted toR2.

Figure 2.12. Plot of an approximate pluricomplex Green function for a 6_−pointed star.

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2.4 On Paper IV

It was shown, in Paper III, that one can approximate the Bergman functions in CN, with probability one, on any determining compact set E ∈ CN. When E is pluriregular and, together with a positive measure µ supported on E, satisfies the Bernstein-Markov property, this leads to approximation of VE.

The construction of a Bergman function of a given order consists of two steps. In the first step, the Gram matrix for a finite family of polynomials is calculated with respect to the inner product in the space L2(E, µ). Then in the second step, an orthonormalization algorithm using the Gram matrix as input, is applied to the original polynomials to convert them to an orthonormal system of polynomials from which the Bergman function is formed.

If the calculations in the first step are based on Monte-Carlo integration, the Gram matrix used as an input of the orthonormalization algorithm is only approximate and hence the result is not exactly an orthonormal set of poly-nomials. The error in the outcome of Mote-Carlo integration can be assessed with the aid of statistics in terms of confidence intervals. However, a question arises, how the resulting errors influence the remaining deterministic calcula-tions.

We can answer this question with the help of the concept of near orthonor-mality. If ε > 0, then linearly independent polynomials q1, . . . , qmare said to

be ε-nearly orthonormal if

k Gram(q1, . . . , qm) − ImkF ≤ ε,

where Imdenotes the (m × m)-identity matrix. Here, Gram(q1, . . . , qm) denotes

the Gram matrix of q1, . . . , qmand k · kF denotes the Frobenius norm.

First, we prove an estimate quantifying how the orthonormalization algo-rithm based on Cholesky decomposition and using a perturbed Gram matrix as an input, produces a nearly orthonormal set of polynomials.

Next, we give a result that describes error resulting from using a nearly orthonormal set of polynomials, instead of an orthonormal one, in calculation of the corresponding Bergman function. To be precise, let k be a positive integer and let E ⊂ CNbe a compact set which is determining for polynomials. Let µ be a positive finite Borel measure supported on E. If 0 < ε < 1 and q1, . . . , qNk is an ε-nearly orthonormal basis forPk(C

N_{) ∩ L}2_{(E, µ) then} Bµ_k(z)(1 − ε) ≤ B(z) ≤ B_kµ(z)(1 + ε), z_{∈ C}N, where B(z) = N_k

∑

j=1 |qj(z)|2, z∈ CN,

is the approximate Bergman function and Bµ_k denotes the exact Bergman func-tion of order k associated with (E, µ).

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3. Sammanfattning på svenska

Denna doktorsavhandling fokuserar på probabilistiska metoder för approx-imation av plurikomplexa Greenfunktioner och bygger på fyra artiklar.

Avhandlingen börjar med en allmän introduktion till användningen av pluri-komplexa Greenfunktioner i flerdimensionell pluri-komplexanalys och en genomgång av huvudegenskaperna. Detta följs av en kort beskrivning av de huvudsakliga resultaten från de bifogade forskningspapper.

I Artikel I studerar vi egenskaper för det metriska rummet av plurireguljära mängder, dvs de kompakta nollställemängderna till kontinuerliga plurikom-plexa Greenfunktioner. Bland de bäst undersökta icke triviala exemplen på sådana mängder finns de sammansatta Juliamängderna, som fås genom itera-tion av ändliga familjer av polynom av flera komplexa variabler. Vi bevisar att det så kallade kaosspelet fungerar bra i fallet. Vi åskådliggör även några sam-mansatta Juliamängder med hjälp av flykttidsfunktionerna och Monte Carlo simulering.

I Artikel II utökar vi resultaten från Artikel I till fallet med oändliga kom-pakta familjer av reguljära polynom av flera komplexa variabler. Med sam-mansättning av funktioner som semigruppoperation genererar vi familjer av oändliga itererade funktionssystem med kompakta attraktorer. Vi visar att sådana attraktorer kan approximeras probabilistiskt på sätt av det klassiska kaosspelet.

I Artikel III studerar vi numerisk approximation och visualisering av pluri-komplexa Greenfunktioner, baserade på Monte Carlo integration. Till skillnad från de alternativa metoder som är beroende av att hitta en följd av noggrant utvalda ändliga mängder som uppfyller några optimala villkor för approxima-tionsändamål, är vårt tillvägagångssätt enklare och mer direkt genom att det beror på slumpmässigt genererade punkter. Vi undersöker numeriskt approx-imationsfelen för vissa enkla geometriska figurer för vilka de plurikomplexa Greenfunktionerna är kända. Om de plurikomplexa Greenfunktionerna inte är kända, så kan felet i Monte Carlo integrationen uppskattas statistiskt med hjälp av konfidensintervall.

Slutligen studerar vi i Artikel IV hur störningar av en ortonomaliserings-process påverkar de resulterande approximativa Bergmanfunktionerna. För

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detta ändamål betraktar vi begreppet av nästan ortonormal hos en ändlig följd vektorer i ett inre produktrum, i betydelsen närheten av Grammatrisen av dessa vektorer till identitetsmatrisen. Vi bevisar uppskattningar av de fel som härrör från att använda nästan ortogonala baser istället för ortogonala. Motivatio-nen för detta arbete kommer från Artikel III. Om Grammatriser beräknas via Monte Carlo integration, så blir resultaten av standard ortogonaliseringsalgo-ritmer nästan ortonormala baser.

Approximation of pluricomplex Green functions: A probabilistic approach

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