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DOI 10.1007/s10958-015-2215-x Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

TOPOLOGY ON POLYNUMBERS AND FRACTALS R. R. Aidagulov and M. V. Shamolin UDC 517.925

For the definition of Julia and Mandelbrot sets, it is necessary to define convergence to infinity of sequences of p olynumb ers (i.e., elements of a finite-dimensional associative algebra over real numb ers, see [5, 6]). The notion of convergence to zero in a vector space allows one to define convergence to any p oint by using a translation. Infinite p oints can b e introduced by compactifications of the space. In the case of p olynumb ers, it is easier to define convergence of a sequence xn to infinity as the convergence to zero of the sequence 1/xn . However, reciprocal values for p olynumb ers are not always defined, so we must also generalize the definition for this case. For the definition of fractals, we also must define admissible functions f (x) whose iterations are calculated for constructing the corresp onding fractal sets. The notion of a converging sequence can b e introduced without any top ology, by the definition of all sequences converging to some p oint. The following natural requirements must b e fulfilled. (1) Filter property. If a sequence xn converges to x and a new sequence yn is obtained from the sequence xn by adding or removing a finite numb er of terms, then the sequence yn also converges to x. (2) Convergence property of a simple ultrafilter. A sequence all of whose terms are equal to x converges to x. Since we consider only finite-dimensional vector spaces over R, it is natural to require the agreement of this definition with the axioms of vector spaces. The space of all sequences can b e equipp ed with the natural structure of a vector space in which the sum of two sequences and the product of a sequence by a numb er are defined comp onentwise. Then the conditions of compatibility with the vector-space structure are as follows: (3) If a sequence xn converges to zero, then the sequence yn = xn + x converges to x. (4) If two sequences converge to zero, then their sum also converges to zero. (5) If a sequence converges to zero, then the product of this sequence with any numb er also converges to zero. To define the convergence structure, it suffices to define a set of sequences converging to zero and satisfying conditions (1), (2), (4), and (5). They form a linear subspace closed with resp ect to the op erations xn yn = xn+1 and xn y n = x
n -1

,

n > 1,

y1 {0, 1},

and are completely defined by these prop erties. In this way, a filtered vector space is defined (sometimes, the term pseudo-topological space is used, see [5]); it is a more general notion than the notion of a top ological space. There is a very large numb er of such spaces. The following requirement that strengthens prop erty (5) is usually added:
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013.

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1072­3374/15/2046­0760 c 2015 Springer Science+Business Media New York


(6) If a sequence xn converges to zero and a sequence of real numb ers n satisfies the condition |n | 1, then the sequence n xn converges to zero. This prop erty is called the star property for filtered spaces. It essentially restricts the numb er of continuous structures. However, these conditions are also sufficiently general and the numb er of such structures of a set grows sup erexp onentially (generally sp eaking, their cardinality is greater than 2Card(X ) ; see [6]). The necessity of a transition to filter spaces that are more general than top ological spaces app ears in the construction of differential calculus on infinite-dimensional vector spaces without norm (see [5]). However, in this case, we can make a shift by filtered spaces that satisfy the following condition: if a set of filters converges to a p oint, then the filter, which is the intersection (as a subset of the set of all subsets of X ) of all of these filters, also converges to this p oint. The numb er of such structures grows only exp onentially as a function of the cardinality of the set X , and top ological structures can b e defined as "coop erations" (see [6]). As in the case of top ological spaces, such spaces can b e introduced using three typ es of axioms (see [2]), namely, by converging filters, by the closure op eration, and by neighb orhoods. However, there are substantial differences b etween these notions in the general case; they are related to the fact that closed sets are not closed absolutely and op en sets are not op en absolutely. We will describ e them in terms of the closure op eration. On the set of subsets of X , we can define a monotonic closure op eration satisfying the following conditions: ¯ (1) the closure A of a set A contains A itself; ¯ ¯ (2) If X Y , then for their closures we have the inclusion X Y (monotonicity of the closure op eration); (3) the closure of the empty set is empty; (4) the closure of the union of sets A and B is the union of their closures. Sets with such closure op erations are called quasi-topological spaces. With such a description, top ological spaces satisfy the following additional axiom: the closure of a closed set coincides with this set, i.e., the closure op eration is the pro jection of the set of all subsets of the top ological space onto the set of closed sets. In contrast to classical top ology, closed sets (closures of certain subsets) in a quasi-top ological space are not quite closed, i.e., generally sp eaking, their closures are larger than they are themselves. Moreover, the pro jectivity condition for the closure op eration is not fulfilled (an op eration (morphism) is called a projector if it coincides with its square). These features also app ear in prop erties of op en sets and neighb orhoods. Nevertheless, these spaces, which are more general than top ological spaces, are more convenient for applications in the theory of calculations with errors, for the construction of the differential calculus in infinite-dimensional spaces (see [5]), and for applications in theoretical physics. The first two axioms are natural for the closure op eration. The third axiom does not significantly affect the theory and is adopted basically for convenience. We explain the fourth axiom. We call spaces satisfying the first three axioms pseudo-topological spaces. Such spaces form a fine continuous structure. The continuity of a mapping f : A B is treated in the ordinary sense: f (X ) f (X ) for any subset X . All these spaces can b e describ ed dually using the op eration of taking the interior; this is similar to the description of top ological spaces by sets of closed sets or by sets of op en sets. The notion of limits is related to the notion of convergence. 761


For this purp ose, we define the category of pseudo-filter spaces in which there is a relation b etween filters and p oints (convergence of a filter to a p oint) satisfying the following conditions: (1) an ultra-filter [x] x converges to x (here and in what follows, we denote by [x] the ultra-filter which is the set of all subsets containing the p oint x); (2) if the prop erty F x holds and F1 is a filter subtler than F , F F1 , then it also converges to x; (3) the ultra-filter [x] converges only to the p oint x. Here the axioms approximately corresp ond to the axioms for pseudo-top ologies. (However, the corresp ondence is not complete since quite different continuous structures are considered.) A continuous mapping is defined as a mapping that transforms converging filters into converging filters; more precisely, a mapping f : A B is said to b e continuous if F x implies f (F ) f (x), where f (F ) is the set of all subsets that contain a certain set f (X ), X F . It is easy to define the structural functor (i.e., the functor that does not change sets) F1 from the category of pseudo-filter spaces in the category of pseudo-top ological spaces: we define the closure of a set X in a pseudo-filter space as the set of all p oints to which the set of all filters that are subtler than the filter [X ] converges. It turns out that the image of this functor does not contain all pseudo-top ological spaces but consists only of quasi-top ological spaces. The definition of continuity using filters approximately corresp onds to the description of top ology by neighb orhoods. In a pseudo-top ological space, we define an op en neighb orhood U of a p oint x as the set that contains x and is the complement of the closure of a certain set. This allows one to define the functor F2 from the category of pseudo-top ological spaces in the category of pseudo-filter spaces as follows: a filter F converges to a p oint x if and only if it is subtler than the filter of all neighb orhoods of the point x. The image of this functor is the category of the quasi-filter spaces satisfying the following additional axiom: (4) The filter which is the intersection (as the set of subsets) of all filters converging to the p oint x also converges to x. Here the fourth axiom does not corresp ond to the fourth axiom of the quasi-top ology. The prop erty of filters themselves is an analog of this axiom: the intersection of two sets from a filter also b elongs to the filter. Is it easy to verify that the restrictions of these functors to quasi-top ological and quasi-filter spaces are mutually inverse. This means that the filter description is another description of quasi-top ological spaces. Fr¨ licher and Bucher also used quasi-top ological spaces for the development of differential calcuoh lus in vector spaces without a norm (see [5]); however, they did not take into account the details mentioned ab ove. The discussion of the need to use more general spaces than top ological spaces for the construction of differential calculus from the p oint of view of category theory requires a separate study; in this pap er, we briefly touch up on this issue. By [6], differential structures form a sub-typ e of continuous structures. A differential structure is defined by the discriminating differentiable functions and by the definition of a pro jection from the set of differentiable functions to the set of p olylinear mappings of tangent spaces. The coincidence of the double and ordinary closures is an algebraic rather than a top ological prop erty; it is used mostly in algebraic top ology. For purely continuous structures, this prop erty is an obstacle for the continuity (in the sense of [6]) of the functors defined; it also creates certain obstacles for the definition of differential structures in spaces without norms. We briefly discuss obstacles created by this prop erty for the definition of continuous structures in the Finsler geometry of Minkowski typ e (with a concave indicatrix). 762


For simplicity, we consider the two-dimensional space in which neighb orhoods of a p oint (x0 ,y0 ) are sets containing the sets |(x - x0 )(y - y0 )| < a, a > 0, for some a. Is it easy to verify that a nonempty set which is a neighb orhood of each its p oints coincides with the whole plane. Therefore, there does not exist a top ological structure (except for the trivial structure containing only two op en sets, the empty set and the space itself ) which is consistent with the Finsler metric of Minkowski typ e. At the same time, these neighb orhoods define a quasi-top ology consistent with the Finsler geometry. Quasi-top ologies are also consistent with roughly rounded structures and structures of fuzzy sets. These structures are not consistent with the top ological prop erty of coincidence of double and single closures. These facts indicate that the restriction by top ological spaces in the definition of continuous structures is unnatural. As we see b elow, the same top ology is not consistent with the continuity of such numb ers in the sets of p olynumb ers with divisors of zero. The set of p olynumb ers is equipp ed with multiplication, which defines an algebraic norm as the determinant of the multiplication matrix that represents an element x. (Recall that p olynumb ers are finite-dimensional associative algebras.) For p olynumb ers, the determinant of the pre-multiplication matrix coincides with the determinant of the p ost-multiplication matrix (they are the determinants of mutually transp ose matrixes). An algebraic norm satisfies the condition |xy | = |x|· |y |. However, there are many functions satisfying only this prop erty. For example, on the set of hyp ercomplex numb ers Hn with a basis consisting of diagonal matrixes we can define the following prop erty: |x| = |x1 |a1 ··· |x|an . In the definition of a continuous structure consistent with the algebraic norm, we can introduce the following semi-norm: x = (|(|x|)|)a , where a is some numb er. The continuous structure itself is indep endent of a, so we choose it from the condition x = ||· x . (1) A semi-norm can also b e defined using a continuous structure. For any vector (p olynumb er) x, we define a p ositive numb er = x such that the sequence 1 , converges to zero. If such a numb er exists for any vector, then the continuous structure is said to b e consistent with the semi-norm. Clearly, the semi-norm of the zero vector is equal to zero. Vectors with semi-norm 1 represent the indicatrix of some (symmetric with resp ect to change of sign) Finsler geometry. A continuous structure in which there exists the corresp onding semi-norm = x defines a top ology consistent with the semi-norm; however, it is p ossible that the continuous structure itself is not defined by any top ology. Now we consider the definitions of the Mandelbrot and Julia sets for commutative p olynumb ers. In this case, the algebra of p olynumb ers itself is the direct sum of several algebras of real and complex numb ers: (2) P = Rk + Cm , xn = y n , y = x, || < where functions are calculated comp onentwise. Therefore, we must examine iterations of the mappings zn
+1

= f (zn ,c)

(3) 763


in algebras of real and complex numb ers. In algebras of real numb ers this process is easily calculated in terms of the b ehavior of orbits. For this purp ose, we calculate stationary p oints (one-p oint orbits) z = f (z, c) and also k -p oint orbits satisfying the condition z=f
(k )

(z, c).

Points for which the numb ers of p oints in the orbit are divisors of k also can b e solutions of the last equation. The other p oints have infinite orbit. Usually, they are contained in an attractor (finite or infinite) of a stationary p oint or a finite cycle, which is defined as a stationary p oint of the kth iteration of the function. This can b e easily illustrated by p olynomial functions. A p olynomial of zero degree has a unique orbit and empty Mandelbrot and Julia sets. A first-degree p olynomial f (x, c) = ax + c for |a| > 1 has a stable stationary p oint at infinity and an unstable stationary p oint c . x = 1-a

(4)

Therefore, the Mandelbrot set in this case consists of the unique zero p oint and the Julia set consists of the unique p oint x . If we consider the set of hyp ercomplex numb ers Hk , we must clarify a continuous structure in this space. We find that the ith comp onent is defined by the formula x
ni

= an (x0i - xi )+ xi . i

(5)

Remark 1. If we introduce the product top ology (which is a Euclidean top ology), sequences of p oints from an orbit tend to infinity if at least one of these comp onents tends to infinity. This occurs when |ai | > 1 or ai = 1 and ci = 0 for at least one of the comp onents. Remark 2. One can try to introduce a top ology with resp ect to a semi-norm assuming that a numerical sequence converges to zero (resp ectively, to infinity) if and only if the semi-norms of terms of the sequence converge to zero (resp ectively, to infinity). In this case, all orbits recede to infinity if the absolute value of the product a1 a2 ... ak is greater than 1 or is equal to 1 but for at least one of the comp onents we have ai = 1, ci = 0. In other cases, the sequence does not tend to infinity. This defines a "top ology" that does not satisfy even the weakest separability axiom. For this "top ology" (we use the quotes since, as a rule, such continuous structures do not corresp ond to any top ology), any function Hk Hm one of whose comp onents is constant is continuous. This "top ology" is the most unnatural for these spaces. In this case, the Mandelbrot set coincides with the whole space if the absolute value of the product of the coefficients is less than 1 or is equal to 1 but there are no comp onents such that ai = 1, ci = 0. In the opp osite case, the Mandelbrot set consists of the union of the hyp erplanes ci = 0 (the set of p oints one of whose coordinates is zero). The Julia set in the first case is the empty set and in the second case is the union of hyp erplanes xi = xi . A detailed analysis shows that this continuous structure is not a top ological space. Remark 3. The most natural continuous structure consistent with the semi-norm is a quasi-top ology whose base of neighb orhoods of zero is defined by the set xi <
xi =0

1 . N

764


Then on sections with one constant (p erhaps, zero) coordinate, neighb orhoods cut the same sets as neighb orhoods in the set Hk-1 . In particular, a sequence in H2 one of whose comp onents is zero converges to some p oint if and only if the other comp onent also converges. In the previous unnatural quasi-top ology, any sequence converges to zero since its semi-norm is zero. The obtained quasi-top ology is not a Hausdorff top ology, but it satisfies the separability axiom T1 . Within the physical framework, p oints p ossessing several (but not all) identical coordinates corresp ond to events b elonging to the same light cone. These p oints were not separable with resp ect to the old semi-norm, however, b ecause of the sp ecificity of the new semi-norm and the corresp onding quasi-top ology, they are separable as in T1 . Finsler spaces of Euclidean typ e are defined by a usual metric and a usual top ology. We show, using the example of p olynumb ers, that quasi-top ologies for Finsler spaces of Minkowski typ e can b e defined by a pseudo-metric describ ed b elow. First, we introduce on Rn a lexicographic order: we assume that (a1 ,... ,an ) > (b1 ,... ,bn ) if aj = bj , j < i, and ai > bi , i n. A pseudo-norm on Hn is a function | ... | : Hn Rn defined comp onentwise as follows: + |h1 ,... ,hn | = |h1 ... hn |1/n max hj
1/(n-1)

,... , | max(hj )| ,

where the ith p osition is occupied by (max(|hj1 ... hjk |))1/k over all sets of k = n + 1 - i different coordinates hj . This pseudo-norm p ossesses the following prop erties. It is nonnegative and vanishes only in the case where all coordinates are zero. It satisfies the inverse triangle inequality (see [1]): (u1 ,... ,un )+ (v1 ,... ,vn ) |u1 ,... ,un | + |(v1 ,... ,vn )| for any two commensurable vectors (i.e., vectors such that ui vi 0 for all i); equality holds if and only if the vectors define the same direction (are prop ortional with a p ositive coefficient in the case of nonzero vectors). We also note that in physics extremums are taken only over commensurable paths (when all tangent vectors are commensurable) and instead of the minimum in Minkowski space one finds the maximum. Thus, the length of an interval of commensurable paths for the case of straight-line paths attains the maximum. The pseudo-norm is also a homogeneous function of degree 1. By the pseudo-norm introduced ab ove, we can define the pseudo-metric (u, v ) = |u - v | with similar prop erties and (pseudo)neighb orhoods in the quasi-top ology. In this case, the Mandelbrot set consists of those c such that for the set of nonzero comp onents ci = 0, the absolute value of the product of the coefficients ai is less than or equal to 1 but among them there is no coefficient such that aj = 1. The Julia set is empty only if the absolute values of all coefficients do not exceed 1 or, in the case where ai = 1, we have ci = 0. If it is nonempty, then it is the union of planes determined by the equations x i1 = x
i
1

,

... ,

xil = xil ,

(6)

so that the absolute value of the product of the coefficients aj that do not b elong to the fixed values (6) does not exceed 1 (if there is 1 among them, then we have the corresp onding relation cj = 0), and the supplement of any aij from the list (6) in the product makes the absolute value of this product greater than or equal to 1. Moreover, the supplemented coefficient satisfies the prop erties aij = 1 and cij = 0. A detailed analysis also shows that this continuous structure cannot b e defined by any top ology since op en neighb orhoods are not neighb orhoods of all their p oints. However, this structure is defined by a quasi-top ology in the sense discussed ab ove. We can show that in p olynumb er sets with divisors 765


of zero, there are no top ologies consistent with semi-norms. The b est continuous structure for is a quasi-top ology of the form sp ecified ab ove. We see that even in the case of a first-degree p olynomial we must clarify the top ology of the and examine a variety of situations typical for higher-order p olynomials. These examples also that the construction of the Julia sets in [6] contains serious errors, and one cannot construct sets only by computer simulations. To explain this situation, we consider the following example with functions of the form x
n+1

them space show Julia in H2 (7)

= 0.6(xn - x )+ x ,

y

n+1

= 2(yn - y )+ y



(we denote by x and y the two comp onents of a p olynumb er). For any C , we can easily calculate the set of p oints (x0 ,y0 ) for which (xn ,yn ) = C . This set consists of branches of hyp erb olas with axes x = x and y = y . This occurs also after the millionth iteration. However, we cannot conclude that the Julia set consists of hyp erb olas. In the case where functions are nonlinear, these sets of hyp erb olas "breed," and the Julia set seems to b e a set with a large numb er of "whiskers." We consider this after examining quadratic functions. In the case of p olynomials of higher degrees, using affine changes of comp onents, one can reduce the problem to the form where the leading coefficient equals 1 while the next coefficient vanishes. For p olynomials of second degree this leads to the following recurrence: x In this case all features of high and unstable stationary p oints. stationary p oint with resp ect to p olynomial function. In the case (8), in addition to
n+1,i

=x

2 n,i

+ ci .

(8)

er-order cases app ear: chaotic tra jectories, limit cycles, and stable Moreover, the infinite p oint (which is a singular p oint) is a stable each comp onent. After examining this case, it is easy to study any

the infinite stationary p oint, we have the following relations: 1 - 1 - 4ci 1+ 1 - 4ci , xi = . (9) xi = 2 2 If ci > 1/4, then the sequence xni tends to infinity superexponential ly for any initial value. If ci = 1/4, then the sequence xni tends to infinity sup erexp onentially except for the case where the initial p oint satisfies the condition |x0i | 1/2. This stationary p oint is unstable from the right and is stable from the left. Therefore, if (10) |x0i | xi ,

then the sequence xni tends to xi , otherwise, it tends to infinity. If ci < 1/4, we have two finite stationary p oints defined by Eqs. (9). The upp er stationary p oint is always unstable (from b oth directions). The lower stationary p oint is stable if and only if 1 3 - ci < . 4 4 In this case, if the strong inequality (10) holds, then the sequence tends to the lower stationary p oint. If an equality holds in (10), then the sequence after the zero term coincides with the upp er unstable stationary p oint. If inequality (10) does not hold, then the sequence always sup erexp onentially tends to infinity. Note that sup erexp onential convergence to zero (to the lower stationary p oint) occurs only in the case ci = 0; therefore, the semi-norm of a numb er can remain b ounded even in the case where several comp onents tend to infinity. In the case of a p olynomial of third or higher degree, this requires convergence to zero with resp ect to at least one coordinate with the same degree. If the inequalities -2 ci < -3/4 hold, the sequence remains b ounded by (10) (if this inequality is fulfilled for the initial value). 766


Now we consider the case where ci < -2. In this case and also in the case where condition (10) does not hold, the sequence tends sup erexp onentially to infinity. However, everything said ab ove holds for the ma jority of initial conditions when condition (10) holds. We introduce the notation I0 = [-xi ,xi ]. Indeed, if x0i I1 = (y1 ,z1 ), -y1 = z1 = -ci - xi , then the next term of the sequence falls outside of the interval I0 and the sequence tends to infinity. Further, for any subinterval Ik (yk ,zk ) for which the k th term falls outside of the interval I0 , we find two symmetric intervals of level (k +1) -ci + yk , -ci + zk , - -ci + zk , - -ci + yk such that if the initial value b elongs to these intervals, then all terms of the sequence up to the kth term inclusively remains in the interval I0 , whereas the (k + 1)th term falls outside of this interval. Therefore, intervals of each subsequent kth level do not intersect with intervals of previous levels and consist of 2k-1 subintervals located inside the complements of intervals of previous levels. Thus, all remaining p oints form a nowhere dense Cantor set in I0 . Having clarified the structure of sequences of iterations of comp onents, we can find the Mandelbrot and Julia sets dep ending on the continuous structure of the space Hk . First, we find the Julia set. As was found ab ove, if ci [-2, 1/4], then the sequence of the ith comp onent remains b ounded, since otherwise this comp onent tends to infinity. Therefore, the product of these intervals b elongs to the Mandelbrot set for any continuous structure. Let us assume that the continuous structure for p olynumb ers satisfies the following condition: if at least one coordinate tends to infinity (sup erexp onentially), then the sequence of p olynumb ers tends to infinity. Then the Mandelbrot set does not have any other p oints. In particular, this is valid for the (Euclidean) product top ology. For strange continuous structures that do not satisfy this condition, we must clarify the notion of the Mandelbrot set. By definition, it is a set of parameters c = (c1 ,... ,ck ) such that the iteration sequence with zero initial values remains in a b ounded domain. In the most unnatural top ology (where b oundedness is understood as the b oundedness of the product of all comp onents, see ab ove), the Mandelbrot set, in addition to the product of intervals, also contains hyp erplanes, where one of the comp onents is identically zero. In continuous structures where coordinates with identical zero are not taken into account, the Mandelbrot set remains the product of these intervals. In particular, this is valid for the natural top ology introduced for the set Hk . Now we define the Julia set. In [3, 4] it is defined as the b oundary of the set J (f ) of initial-value p oints for which the sequence tends to infinity. However, this definition is not quite accurate. From the meaning of the unstability of the dynamics of iterations and in agreement with the definition of the Mandelbrot set and the relation b etween these sets, a more exact definition is as follows: J (f ) is the boundary of the sets of initial-value points when the sequence of iterations remains bounded. According to this definition, we do not need the definition of convergence to infinity. The difference b etween these definitions is that in the second definition the Julia set is less by a set of initial-value p oints for which the sequence of iterations is unb ounded but does not tend to infinity. In the classical case of the complex plane and rational functions, these definitions coincide. They also coincide in the case of natural quasi-top ologies on Hk for rational functions; however, due to the choice of "bad" functions or an unnatural quasi-top ology, they may differ by the set sp ecified ab ove. Obviously, we need to clarify the notion of b oundary. In natural situations, the set J (f ) is op en, so that its complement (the Fatou set) is closed, and the notion of the b oundary in these condition is irrelevant to the definition of the Julia set. However, there 767


exist cases where the prop erties sp ecified ab ove are not fulfilled and, p erhaps, some clarifications in the definition of the Julia set in terms of b oundary are needed: it must b e defined using the op erator of (unnatural) closure of the set itself and its closure. In the case of the Euclidean top ology, where for at least one comp onent the inequality ci > 1/4 holds, the set J (f ) coincides with the whole space and hence the Julia set is empty. If, moreover, none of the coefficients vanishes (cj = 0), then the Julia set is also empty for the unnatural and natural quasi-top ologies compatible with the semi-norm considered ab ove. We denote by Ji = [-xi ,xi ] either the set of p oints for -2 ci 1/4 or the Cantor set constructed ab ove for ci < -2. In the Euclidean top ology, the Julia set coincides with the b oundary of the product J = Ji of these sets. The b oundary of the product is defined traditionally as J =
i i

Ji â
j =i

Jj

(here the sum of sets is meant as the union); moreover, the b oundary of the Cantor set coincides with itself and the b oundary of an interval consists of two p oints. This notation can b e extended to the case where certain coefficients satisfy the inequalities ci > 1/4 and Ji = (the product of a set and the empty set is empty and the b oundary of the empty set is empty). In the case where none of the coefficients ci vanishes, the Julia sets are the same as in the case of the Euclidean top ology, since no exp onential convergence whatsoever can comp ensate sup erexp onential divergence at infinity with resp ect to at least one coordinate. If there are zero coefficients in top ologies related to a semi-norm, additional subsets of the Julia set besides the set specified above appear. To illustrate the app earance of additional branches in the Julia set in this case we consider the following example in H2 : x
n+1

= x2 , n

y
n

n+1

2 = yn + c.

The first recurrence can b e easily solved: xn = x2 . 0 It is easy to verify that for any initial value y0 that does not b elong to the set J2 corresp onding to the y -comp onent, the following inequality holds for some combinations of parameters: a(y0 ) = exp
n

lim

ln(yn ) > 1. 2n

This function is continuous outside the interval [-y ,y ]. Therefore, if |x0 |a(y0 ) < 1, then the seminorm tends to zero. If the opp osite inequality holds, then the semi-norm tends to infinity. Hence, the set of p oints |x0 |a(y0 ) = 1 similar to a hyp erb ola is added to the Julia set. In the case of the natural top ology, we have the set whose first coordinate is zero and whose second coordinate runs through the complement to the set J2 . More general cases of p olynomials of higher degrees can b e considered similarly. Now we explain the failure of the computer simulation used in [4] for the construction of the Julia set. In [4], the cases c = -1.3+ 0 · j (c1 = c2 = -1.3) and c = -1+ 0.2 · j (c1 = -0.8, c2 = -1.2) were considered, i.e., the cases where -2 < ci < -3/4 and the orbits remain b ounded in the interval [-xi ,xi ] if the initial values are contained in this interval or sup erexp onentially tend to infinity otherwise. One of the comp onents, b eing b ounded in the interval [-xi ,xi ], p erforms a chaotic motion inside this interval since, due to the instability of the lower and upp er stationary p oints, the inequality ci < -3/4 holds. 768


Fig. 1 In the case where the initial value satisfies the equation f
(k )

Fig. 2

(x) = f f

(k -1)

(x) = 0,

f

(1)

(x) = f (x) = x2 + c,

p ossessing 2k real roots from the corresp onding interval, the k th term is equal to 0. If the initial value is very close to one of these solutions, then the kth term is close to zero and, therefore, the semi-norm of the k th term can b e small and the impression is created that the terms do not tend to infinity. Such values exist for any numb er. The main mistake here is that the authors did not calculate the norm of the (k + 1)th term, which is large for large values of k. Therefore, these p oints must b e excluded from the list of p oints with b ounded orbits. We also note that if the coefficients satisfy the condition g
(k )

(c) = 0,

g (c) = c2 + c,

then for this countable numb er of values of the coefficients, the values of the comp onents p eriodically vanish; examining the norms of terms of the iterative sequence of p olynumb ers, we will b e able to guess at the b oundedness of the sequence only after a sufficiently large numb er of steps. Moreover, tra jectories (orbits) obtained for these sp ecial values of coefficients are sup erstable kcycles, i.e., in the case where an initial p oint is sufficiently close to a p oint of the cycle, the orbit sup erexp onentially tends to this cycle passing through the zero p oint. This leads to the fact that for these sp ecial values of coefficients, the Julia sets presented in [4] actually differ from the Julia sets defined by us more precisely by meaning (see [7­27]. We draw the domains defined by the following inequalities: I
nk

= {(x0 ,y0 ) : |xn yn | < k A, |x

n+1 yn+1

| < k A}, A = x y .

For small 0 < k < 1, the domains can b e not simply-connected and they are not imp ortant for separation of the domain of initial values for which the iteration sequence (orbit) is b ounded from the domain of initial values for which the iteration sequence is unb ounded. In Figs. 1­?? (here c1 = c2 = -1.3 and n = 1, 2, 3, 4, 5), we illustrate the cases where k = 1, 2, 3, and 4. These figures show that the "tails" are cut off. If an infinite tail exists for the domain |xn yn | < k A, it is cut off by the b oundary of the domain |xn+1 yn+1 | < k A and vice versa. Thus, the qualitative b ehavior is p ossible in the case where -2 c1 ,c2 < 1/4. These domains have finite (cut off ) tails and the domains themselves converge to the rectangles |x0 | < x , |y0 | < y . Acknowledgment. This work was partially supp orted by the Russian Foundation for Basic Research (Pro ject No. 12-01-00020-a). 769


Fig. 3

Fig. 4

Fig. 5

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