Attractors of Compactly Generated Semigroups of Regular Polynomial Mappings
We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.
In the recent paper  it was shown, as a part of the investigation of the space of pluriregular sets, that it is possible to approximate composite Julia sets generated by finite families of proper polynomial mappings in in a probabilistic manner. This can be done in the spirit of the theory of iterated function systems (IFSs) and the so-called chaos game. The aim of this paper is to prove similar results in the case of infinite compact families of polynomial mappings. Inevitably, the topological and probabilistic aspects get more complicated than those in the finite case. The main motivation for this study is the wish to gain a better understanding of the metric space of compact, pluriregular, and polynomially convex subsets of . This, however, requires a very careful analysis of different types of Julia-like sets arising naturally in this context. This variety of Julia sets is easier to grasp, if one looks at them as corresponding to the topological semigroup generated by infinite compact families of proper polynomial mappings. This is consistent with the point of view adopted by a number of researchers in one complex variable (see  and, e.g., the work of Stankewitz and Sumi [3–5]).
As a visual hint of the additional complexity that infinite families bring about, we can consider what happens in the complex plane when instead of inspecting the filled-in Julia set of a single polynomial, in this case with , we examine the filled-in Julia set generated by the compact infinite family of polynomials , where is a closed-square centered at . In the following pictures, and . Figure 1 shows the autonomous Julia sets of the polynomials , one with and the other eleven with selected at random from according to the uniform probability distribution.
Bearing in mind that this is just a tiny selection of Julia sets of the simplest (autonomous) type, one can appreciate the infinite variety of Julia sets (autonomous or not) that can be obtained by using just this family of simple quadratic polynomials. The union of all these sets would constitute the composite nonautonomous Julia set corresponding to all combinations of . An approximate outline of this set is depicted in Figure 2, to the right of the filled-in Julia set for included for comparison. All these sets were plotted with the help of measuring the escape time of the orbits of the points under the iteration process. The shades of grey mark how quickly the considered orbits go beyond the radius of escape. Obviously, the situation gets more involved with more complicated polynomial mappings and in higher dimensions.
The paper is divided into seven sections including the introduction.
In Section 2, we take a closer look at the nature of convergence in the compact-open topology in the space of polynomial mappings in and, in particular, at the link to the coefficients of such mappings and their compositions. We also recall the definition of regular polynomial mappings and the concept of radius of escape and its basic properties. Moreover, we propose to regard the topological semigroups generated by compact families of regular mappings, with the composition of mapping as the semigroup operation, as the principal objects that give rise to the composite Julia sets that we want to study.
In Section 3, we recall the definition of the pluricomplex Green function of a nonempty compact subset of and the concept of pluriregularity. We also review the definition of the metric space of all polynomially convex pluriregular compact sets in . The Julia sets we are studying in this paper reside precisely in that space. The space is known to be complete (see ), and separable (see ) but not proper, in the sense that bounded closed sets do not have to be compact (see ). The topology of still holds many unanswered questions. We discuss in some detail the intricacies of the closure operation in . Namely, given a subset of a subset of , we compare the closure of the union in of the sets which are elements of with the union in of the sets which are elements of the closure of in . It turns out that equality between these sets requires additional assumptions.
In Section 4, we prove that if is a regular polynomial mapping, then the contraction is a similitude, which is also continuous when regarded as a function of two variables . We also show that if is a compact family of regular polynomial mappings of a fixed degree and is compact, then is also compact. Furthermore, is a contraction whose fixed point can be described as the atlas of the Julia sets generated by sequences from the topological semigroup generated by . We could also describe the set as the attractor of the infinite iterated function system .
Section 5 begins with restating the definitions of autonomous filled-in Julia set generated by a single regular polynomial mapping and a nonautonomous filled-in Julia set generated by a sequence of regular mappings. If the sequence comes from a compact family of regular mappings with a fixed degree, then we show that converges to as . Moreover, the speed of the convergence can be estimated in terms of the natural metric on . We also furnish the code space with a metric like the one used in the classical case when the family is finite. We close this section by linking the attractor to other types of Julia sets. Namely, consists of all sets with and the union of all sets constituting is the partly filled-in composite Julia set generated by , whereas the polynomially convex hull of is the filled-in composite Julia set generated by . We also include some comments to justify the use of semigroup terminology in this context.
The last two sections contain a counterpart of Theorem 2 in  in the case of a compact infinite family of regular polynomial mappings of the same degree. Section 6 presents an extension of Theorems 2(a) and 2(b) from . Essentially, we show how much the attractor pulls iterations of sets from the surrounding space towards itself if the polynomial mappings used in the iteration process come from . In Section 7 we extend Theorem 2(c) from  to the case of compact infinite families and we prove that the chaos game approximation of the partly filled-in composite Julia sets remains also valid in this case. We describe first the deterministic version based on disjunctive sequences and then the more familiar probabilistic version. Finally, we close the article with a few comments linking the mathematical context we have investigated to the study of invariant measures associated with general probabilistic approach to iteration function systems described in .
A few words about the notation used in this paper are in order. For any nonempty sets and , the symbol will denote the set of all functions from to . If is a collection of nonempty subsets of a set , the symbol will always denote . Let be a metric space. The symbol will denote the set of all nonempty compact subsets of ; will denote the open ball with center and radius whereas , , and will denote the distance of a point from a set, the diameter of a set, and the Hausdorff distance between two compact sets, respectively. The set , where , will be referred to as the -dilation of the set . In the case of the Euclidean metric in , we will drop the subscript . A norm symbol with a subscript will always denote the supremum norm. We will use the convention that stands for nonnegative integers and for natural numbers (excluding zero). Other notational conventions will be described later as the need for them arises.
2. Semigroups of Regular Polynomial Mappings
If , then by we denote the vector space of all polynomial mappings of degree not greater than . Since is of finite dimension, all norms defined on it are equivalent. In particular, if is compact and determining for polynomials (i.e., is not contained in the zero set of a nonconstant polynomial), then a natural choice is the supremum norm , where , and is endowed with the Euclidean norm. Another natural choice would be to transfer the norm from the Euclidean space of Taylor’s coefficients using the natural isomorphism: where the multi-indices in are ordered according to the graded lexicographic order and
When is the closed polydisc with the center at the origin and radius , then we can use Cauchy’s estimates to establish a quantitative link between these two norms. For any , if , with some , then we have the following:
Consequently, the topology on is the topology of uniform convergence of polynomial mappings on compact sets or, equivalently, the topology of convergence of the coefficients of polynomial mappings. To put it differently, it is the topology induced on from the set of all polynomial mappings furnished with the compact-open topology, that is, the smallest topology containing all the sets of the form , where is compact and is open. The following statement will be useful later on.
Proposition 1. The composition mapping is continuous. In fact, if polynomial mappings are identified with their ordered sets of coefficients, then the mapping is a polynomial mapping between the respective spaces of coefficients.
Proof 1. It suffices to consider .
The first statement can be checked directly on the sets from the neighbourhood subbase of the topology of . If , then for some compact set , we can have the inclusions and . This means that is contained in the inverse image of under the composition mapping, which completes the proof of continuity.
As for the second statement, in view of (1) and (3) it is enough to observe that the mapping is a polynomial.
If , we will denote by the homogeneous component of of degree . We say that is regular if . The subset of all regular maps in , denoted by , is an open subset of (see Section 2 of ). Regular polynomial mappings are proper (cf. , Theorem 5.3.1) and so they are closed. As proper holomorphic mappings, they are also open and hence surjective (see , p. 301).
Throughout this paper, will denote the closed Euclidean ball in with center at the origin and radius . If , then .
In what follows, let denote the th iterate of , that is, the composition of copies of . We call an escape radius for , if for every , we have
Note that if is an escape radius for , then all numbers bigger than are also escape radii for the same mapping.
In  (Lemma 1), it was proved that there exists a continuous function, such that (given by a constructive formula) is an escape radius for . Another useful observation is that if , then (cf. , Lemma 1).
In our investigation, we will consider a nonempty compact subset of . It is worth mentioning that such a family is regular in the sense defined in . Indeed, a subset of is regular there if and only if it is relatively compact in . One simple example of such a compact family was already mentioned in the Introduction section.
If the composition of mappings is the semigroup operation, then because of Proposition 1, any nonempty compact subfamily of generates a topological semigroup denoted by , which in turn can naturally be associated with a Julia-type set. The primary objective of this article is to investigate such Julia sets and, more specifically, the approximation of such sets. The reason for invoking the concept of a semigroup in this context will be explained at the end of Section 5.
3. The Space of Pluriregular Sets
If is a nonempty compact subset of , its pluricomplex Green function will be denoted by . For the background, we refer the reader to . Recall that where is the Siciak extremal functionwith the supremum being taken over all nonconstant complex polynomials such that . It is easy to check that for any compact set , the zero set of is equal to the polynomially convex hull of . A compact set is said to be pluriregular if is continuous.
Let be the family of all compact, pluriregular, and polynomially convex subsets of . Endowed with metric defined by turns out to be a complete metric space (see Theorem 1 in ). It is worth observing that the above formula defining can also be used for pluriregular sets and which are not necessarily polynomially convex. In this case, we obtain a pseudometric on the set of all pluriregular compact subsets of . Note also that if , and is a set such that , then .
It was shown in Theorem 1(a) from  that if is compact in , then is compact, being bounded and closed. In contrast, according to Theorem 1(d) in , a closed and bounded set in does not need to be compact, since the space is not proper. In connection with these results, we would like to address here two questions, the answers to which can facilitate a better understanding of the topology of space .
The first question concerns the operations of closure in and in . Let . Is it true that
It turns out that the answer depends on these additional assumptions: (i)It is affirmative, if is compact in . Indeed, , where the second equality follows from Theorem 1(a) in .(ii)However, the equality (13) is not true in the general case. To be more precise, we have the following properties: (1)If is relatively compact in , the inclusion “” in (13) holds. Namely, is closed by Theorem 1(a) in  and the inclusion follows from (2)The inclusion “” in (13) does not hold in general, even for a relatively compact set . To see this, consider the following example from Section 3 in . Take and . We have (3)If is not relatively compact, the inclusion “” in (13) does not need to hold either. To see this, recall Example 3.6 from . Take with where is so small that with cap denoting the logarithmic capacity. There exists such that On the other hand, if , then there exists with , which means that we can find a subsequence such that as . At the same time, Therefore, in this case, as . Thus, . Hence,
The other question concerns the fact that in the compactness of a subset is equivalent to being closed and bounded, but it is not the case in . It is natural to ask whether compactness is needed in the assumption of Theorem 1(a) in  mentioned earlier. Let be closed and bounded in . Does have to be compact in ? The answer is no, it does not. Take and from point (3) above (we use once again Example 3.6 in ). Since , there exists a sequence with . Since and , the set is not closed.
4. Similitudes of the Space of Pluriregular Sets
Let us recall the transformation formula for regular polynomial mappings from Theorem 5.3.1 in :
Recall also that if is a metric space and is a constant, then a mapping is referred to as a similitude with the ratio , if for all . As a direct consequence of (20), we can describe a family of similitudes of .
Proposition 2. If , then is a contractive similitude with the contraction ratio .
Proof 2. Let . In view of (20) we have
And this concludes the proof.
Proposition 3. The mapping is continuous with respect to the product topology on .
Proof 3. Fix and . In view of the triangle inequality and Proposition 2, if , then
Hence, it is now enough to prove that if , as , then
Take a sequence which is convergent to and consider . This family is compact in and therefore, is bounded in view of Remark 3.2 in , because . Take such that . Since is a norm in , there exists such that for . This means that for such . Obviously, , too.
Fix . The Green function is continuous; hence, it is uniformly continuous on , that is, there exists such that if with , then . Since , there exists such that if . Therefore,
And this concludes the proof.
Let now be a compact subset of . For any subset of , put where the similitudes are as in Proposition 2.
Proposition 4. If is a compact subset of and is a compact subset of , then is compact.
Proof 4. Choose a sequence of elements from . Then, there exist sequences and such that . As is compact, we can assume (passing to a subsequence if needed) that in if . Since is compact, so here again (passing to a subsequence if needed), we can assume that in if . It follows from Proposition 3 that , if . Thus, we have shown that every sequence in has a convergent subsequence.
Recall that denotes the family of all nonempty compact subsets of the metric space , furnished with the Hausdorff metric.
Corollary 1. Let be a nonempty compact subset of . The mapping is well defined and is a contraction with ratio . In particular, the mapping has a unique fixed point .
5. Julia-Type Sets
If , its (autonomous) filled-in Julia set is defined as follows:
As shown in , this set is the unique fixed point of the similitude . Hence, the standard argument used to prove the Banach contraction principle yields the equality
Moreover, if is an escape radius of , then we also have the equality
Before turning our attention to other types of Julia sets, we need to point some useful estimates. If is an escape radius for , then (cf. Equation 7 in )
More generally, if is a compact family in , then due to the continuity of the mapping in (8), a common escape radius for all mappings in can be found. Also, is finite because of the compactness of . Thus, as an immediate consequence of (34) we obtain
For a sequence of mappings from , we define its filled-in Julia set (nonautonomous if the sequence is not constant) as follows:
The estimate (36) allows the use of the enhanced version of Banach’s contraction principle (Lemma 4.5 in ) for sequence . As a consequence, we can see that if is as in (36). For some background on (a larger family of) nonautonomous Julia sets in the complex plane, see [14, 15].
It turns out that nonautonomous filled-in Julia sets can be approximated by autonomous filled-in Julia sets. Before making this statement more precise, let us establish some notations. If is a compact family in , the symbol will denote the code space over , defined as the Cartesian product of countably many copies of with the usual product topology. By Tychonoff’s theorem, is compact and it can be furnished with the metric (see, e.g., Theorem 4.2.2 in ):
Proposition 5. Let be a compact family in . Then, for each and where . In particular,
Proof 6. To show (40) one can repeat the proof of the enhanced version of Banach’s contraction principle (Lemma 4.5 in ). Namely, in view of (36), we have
Letting go to infinity gives (40).
As for (41), in view of (40) we can write
We define the partly filled-in composite Julia set of the compact family as
This set is compact (see proof of Theorem 4.6 in ), and its polynomially convex hull is the unique fixed point of the mapping: is called the filled-in composite Julia set of . Here, the hat marks the operation of taking the polynomially convex hull of the set under the hat. The subscript tr stands for the word truncated.
The following theorem describes the connection between the Julia sets from this section and the attractor from the end of the previous section (Corollary 1).
Theorem 1. Let be a nonempty compact family in . Then,(1) (2) .
The family is the unique fixed point of (cf. Corollary 1). Therefore, Since by Proposition 2 the function is a contraction, The sequence is also decreasing with respect to inclusion. Therefore, its limit is a singleton, and by the definition of , we have the equality Thus, .
From the definition of , it is obvious that . Let us fix a common escape radius for all .
Now, take . First, we claim that for any , there exists contained in the -dilation of and such that . Indeed, given , one can choose so that the inequality is satisfied with defined in Proposition 5. Since , by the definition of the latter, there exist such that The inequalities (49) and (40) imply the estimate and this means that fulfills our claim.
To finish the proof, we want to show that . Since is compact, the sequence has an accumulation point . But then, since convergence of sets in means uniform convergence of the corresponding pluricomplex Green functions, we can conclude that , which means that .
Remark 1. It is worth emphasizing that all of the types of Julia sets defined in this section correspond one way or another to sequences in the semigroup . This is the reason why conceptually it is natural to see the set not only as the attractor associated with the semigroup but also as a kind of atlas of all Julia sets associated with that semigroup. Indeed, this is exactly the meaning of Theorem 2 combined with the definition of .
6. On the Attracting Nature of
Recall that we use the symbol to denote the open ball in with center at and radius .
Theorem 2. Let be a nonempty compact family in .(a)Let . If and is an open subset of , then almost all elements of the sequence belong to . In particular, all accumulation points of this sequence are in and so is compact in .(b)Let . For every neighbourhood of , there exists an open set and mappings , such that Moreover, can be made arbitrarily large.
Proof 8. Fix a common escape radius for all . Let be like in Proposition 5. Fix . In view of the proof of (40) (but with replacing ) combined with the triangle inequality and (36), we have the following estimates:
which is what is needed, as .
Take and such that . Fix . Without loss of generality, we may suppose that It follows from Theorem 1 that for some . Moreover, by (41). Therefore, we can choose such that
Define for and let
If , then there exists such that . Therefore, Moreover,
7. Chaos Game and Approximation of Attractors
We will start with the definition of disjunctive sequences over a finite or countable alphabet.
Let be a nonempty set which is at most countable. A sequence of elements of , that is, a function is said to be disjunctive, if for any and any function there exists such that for . A simple example of a disjunctive sequence with is given in : the first entry is 1, followed by all 2-letter words over , then by all 3-letter words over , and so on.
If is regarded as the alphabet and functions like as possible finite words over , then the sequence is disjunctive if it contains all finite words as its finite subsequences. Disjunctive sequences, usually over a finite alphabet, have been used for a long time in study of formal languages, in automata theory and number theory (see  for an overview). More recently, disjunctive sequences turned out to be a natural tool for derandomization of the chaos game (see ).
Theorem 3. Let be a nonempty compact subset of and a dense countable subset of . Let be a disjunctive sequence.
Then, for any , where
Proof 9. First of all, it should be noted that Theorem 2 yields compactness of . Furthermore, a countable subset of exists because of the separability of . Recall also that is separable (see Theorem 1(d) in ).
Fix a norm in .
In view of Theorem 2(b) and from Theorem 1(b) in , it suffices to prove that where denotes the Hausdorff metric corresponding to .
Take . In view of Theorem 2(a), if is sufficiently large, then the -dilation of contains , and hence also . In order to prove that for sufficiently large , the -dilation of contains , it is enough to show that any point from an -dense finite subset of is within -distance from a point of .
Let be an element of a fixed -dense finite subset of . By Theorem 2(b), there exist and such that for the image of the -dilation of via the mapping is a subset of . Using Theorem 2(a) again if necessary, we can increase so that the -dilation of contains . In particular, if , then
Since is dense in , there exist such that . Let be chosen so that for .
Since is disjunctive, for some , we have for . Consequently, if we put , we have and
We know that because of the choice of , and so it follows from (63) combined with the triangle inequality that
And this concludes the proof.
The next statement is a probabilistic version of the above theorem.
Corollary 2. Let be a nonempty compact subset of and a dense countable subset of . Let be generated according to probabilities such that , that is, the values of are chosen at random, independent from each other, so that for .
Then, for any , with probability 1, where
Proof 10. Because of the strong law of large numbers applied to Bernoulli processes, we can conclude that, given a finite word over the alphabet , the sequence contains this word with probability 1. Hence, we can use the same reasoning as in the theorem above.
We would like to finish the article with a general observation.
Let us assume that we have a probability measure on some -algebra of subsets of , where as in Theorem 3, is a compact subset of . We will follow the general set-up from . We will be concerned with a Markov chain , with initial state and where are independently and identically distributed (IID) random elements in with probability distribution . Let also denote the induced probability measure on the code space .
In a more general setting, the initial state can be given by a random element in , independent of , and with the probability distribution . Then, it is natural to define the random elements
So in particular, is the probability distribution of . If we also define as the probability distribution of , then the probability distribution of is .
The reverse order chain is defined to be as follows:
Because of the IID property, both and have the same probability distribution
Note that all of the above definitions make sense because Proposition 3 is guaranteeing appropriate measurability of the sets.
Let denote the Dirac measure concentrated at , that is, . Below, we use the mapping
It is continuous because of the estimate (40) combined with the definition (39) of the metric on . Indeed, given and , choose so that . If is such that , then and thus in view of (40) combined with the triangle inequality.
Proposition 6. Let be a nonempty compact subset of , let be a probability measure on some -algebra of subsets of and let also denote the induced probability measure on .
Let be the pushforward measure on obtained from the measure on the code space via the mapping . Then: (a)If is a Borel probability measure, then weakly. In particular, and is the unique probability measure invariant with respect to .(b)For all and for a.e. weakly.(c)The support of is ; hence, this is the unique fixed point of the iterated function system . In particular, the support of is compact.
Proof 11. (a) and (b) are straightforward consequences of  (Theorem 8).
(c) By Theorem 8 (15) from , there exists , which may depend on and , such that Therefore, . On the other hand, and hence .
It should be noted that the novel element in the above observation is the compactness of the support of the measure in the case of infinite family and its invariance under the IFS in this case. Theorem 8 in  gives this property, but only in the case of finite iterated function systems.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The research of the third author was partially supported by the NCN Grant no. 2013/11/B/ST1/03693 and that author is also grateful to Uppsala University for its hospitality. The authors wish to thank Margaret Stawiska-Friedland and the anonymous referees for their helpful remarks.
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