aa r X i v : . [ m a t h . D S ] A ug PLANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIASETS
YURY KOCHETKOV
Abstract.
A tree, embedded into plane, is a dessin d’enfant and its Belyifunction is a polynomial — Shabat polynomial. Zapponi form of this polyno-mial is unique, so we can correspond to an embedded tree the Julia set of itsShabat-Zapponi polynomial. In this purely experimental work we study rela-tions between the form of a tree and properties (form, connectedness, Hausdorffdimension) of its Julia set. Introduction
Shabat polynomial of a plane bipartite tree is not unique, but we can made itunique, if we demand that: a) critical values are +1 and −
1; b) sum of coordinatesof white vertices (i.e. inverse images of 1) is 1; c) sum of coordinates of blackvertices (i.e. inverse images of −
1) is −
1. Shabat polynomial with these propertieswill be called Shabat polynomial in Zapponi form, or Shabat-Zapponi polynomial,or SZ-polynomial [7]. Thus, we can correspond to a tree the Julia set, i.e. the Juliaset of its SZ-polynomial. We want to understand is there a correspondence betweengeometry of a plane tree and such properties of its Julia set as form, connectednessand Hausdorff dimension?
Remark . At first it was expected that Julia set of a Shabat polynomial is some-thing simple with Hausdorff dimension approximately 1 (because Shabat polyno-mial is a generalized Chebyshev polynomial). This assumption turned out to bewrong. So, we decided to study the Zapponi form of Shabat polynomial, becauseif there exists a SZ-polynomial for a given bipartite tree, then such polynomial isunique.In the course of this experimental work we found that: a) there is some similaritybetween the form of a given tree and the form of its Julia set; b) the connectednessof Julia set is probably the main characteristic of an embedded tree.2.
Definitions and notations
Zapponi form of Shabat polynomials and its properties.
We considerplane bipatite trees, i.e. trees embedded into plane, with vertices properly coloredin black and white. A polynomial p with exactly two finite critical values — one andminus one will be called Shabat polynomial [5]. The inverse image T ( p ) = p − [ − , − ,
1] is a plane bipartite tree, where white vertices are images of 1 andblack — of −
1. For each plane bipartite tree T there exists a Shabat polynomial p such that trees T and T ( p ) are isotopic. Such polynomial will be called a Shabatpolynomial of the tree T . If polynomials p and q are Shabat polynomials of thesame tree T , then q ( z ) = p ( αz + β ) for some constants α = 0 and β . A Shabat polynomial is in Zapponi form [7], if the sum of coordinates of whitevertices is 1 and black vertices — − Proposition 2.1.
Let T be a bipartite tree and p = a n z n + a n − z n − + . . . + a z + a — its SZ-polynomial. Then a n − = 0 .Proof. Let x , . . . , x s be roots of polynomial p − k , . . . , k s ,respectively, and y , . . . , y t , l , . . . , l t be roots of p + 1 and their multiplicities. Then s X i =1 k i x i = − a n − a n = t X j =1 l j y j ⇒ s X i =1 k i x i + t X j =1 l j y j = − a n − a n . (1)Also we have, that p ′ ( z ) = na n z n − + ( n − a n − z n − + . . . + a = na n s Y i =1 ( z − x i ) k i − t Y j =1 ( z − y j ) l j − . Hence, s X i =1 k i x i + t X j =1 l j y j = s X i =1 k i x i − s X i =1 x i + t X j =1 l j y j − t X j =1 y j = − · ( n − a n − na n . (2)From (1) and (2) we have that a n − = 0. (cid:3) Corollary 2.1.
If p is a Shabat polynomial and a n − = 0, then s X i =1 x i = − t X j =1 y j . Corollary 2.2.
Let T be a bipartite tree. If there exist its SZ-polynomial p , then p is unique and its field of definition coincides with the field of definition of the tree T [5]. Proof. If p = a n z n + a n − z n − + . . . + a is a Shabat polynomial of a tree T and a n − = 0, then p is unique up to variable change z := αz and the unique choice of α in this variable change gives us SZ-polynomial.Let now K be the field of definition of a tree T and q = b n z n + . . . + b ∈ K [ z ] beits Shabat polynomial. The variable change z := z − b n − /b n preserves the fieldof definition, but turns coefficient at z n − to zero. If X = P x i , then X ∈ K . If X = 0, then the the variable change z := X · z also preserves the field of definition,but turns (in new coordinates) X to one. Then Y = P y j = −
1. If X = 0, thenShabat polynomial in Zapponi form does not exist for the tree T . (cid:3) Remark . In [7] it was proved that SZ-polynomial always exists for trees withprime number of edges. SZ-polynomial obviously does not exist, if the tree issymmetric, i.e. if it has a nontrivial rotation automorphism with the center in oneof vertices.
Conjecture.
SZ-polynomial exists for non-symmetric trees.
In what follows SZ-polynomial for a tree T will be denoted p T . LANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIA SETS 3
Julia sets and Hausdorff dimension.
Definitions of Fatou and Julia setssee, for example, in the book [6]. For us the following properties of Julia sets willbe important. • Julia set of a polynomial p is the boundary of the basin of infinity, i.e. theboundary of open set of those points, whose iterations converge to infinity. • Let A be the set of stationary repelling points of p and let A i = p − ( A i − ), i >
0. Then Julia set of p is the closure of ∪ i A i . • Julia set of p is connected if and only if iterations of critical points of p constitute a bounded set. In the case of a SZ-polynomial it means thatconnectedness of Julia set is equivalent to the boundedness of iterations of1 and −
1. If iterations of 1 and − − Remark . It must be noted that performance of these algorithms differs from caseto case. Box counting method does not work, if Julia set is totally disconnected.It also demonstrate bad performance, if Hausdorff dimension of Julia set is > . Julia sets of SZ-polynomials.
Let T be a bipartite tree and let T be thesame tree, but with inverse colors (i.e. white vertices in T are black in T and blackvertices in T are white in T ). Then p T ( z ) = − p T ( − z ). Let a be an arbitrarypoint and p T ( a ) = a , p T ( a ) = a , p T ( a ) = a and so on. Then p T ( − a ) = − a , p T ( − a ) = − a , p T ( − a ) = − a , and so on. It means that Julia sets of polynomials p T and p T are the same up to rotation on π around the origin, i.e. characteristicsof Julia set depends only on tree and not on its coloring. In what follows we willstudy one tree from the pair ( T, T ). Remark . Let T be a bipartite tree. By fixing some white vertex of degree > > − p of T . In this case p will be a postcritically finite polynomial (a pcf-polynomial),i.e. a polynomial with finite orbit of set of critical points (see [1]). It must be notedthat Shabat pcf-polynomial of a tree T is not unique. Example 2.1.
Let T be a tree with four edges: rr ❝ r ❝ (cid:0)(cid:0)❅❅ Then p = − ( z + 1) (3 z − − p T = 2(2 z + 1) (2 z − YURY KOCHETKOV is its SZ-polynomial. Julia sets of p and p T are quite different:Figure 1. Figure 2.Julia set of pcf-polynomial p . Julia set of SZ-polynomial p T .In the left figure iterations of yellow points converge to infinity, of green points —to −
1, of red points — to 1. Julia set is connected. Its Hausdorff dimension approx-imately equals 1 .
17 (box counting method) or 1 .
13 (packing dimension method).SZ-polynomial p T has a weakly attracting 10-cycle. Let O be a union of the domain { z | abs( z ) > } and 0 . O in 5 steps or less are white, in 6 or 7 steps — green, in 8, 9 or10 steps — red. All other points (including points of Julia set) are blue. Julia setis connected. For its Hausdorff dimension box counting method gives estimation ≈ .
62, packing dimension method — ≈ .
35, JP-algorithm — ≈ . General remarks
Let T be a tree and p T — its SZ-polynomial. Characteristics of Julia set J ( p T )depend on behavior of iterations of ±
1. There are several types of this behavior.3.1. ”Generic” types.g1:
Iterations on ± p . Here Julia set is acommon border of two basins: the basin B ∞ of infinity and the basin B p ofattracting point p . As all vertices of T belong to B p , then the form of Juliaset resembles the form of the tree (in some general manner). Hausdorffdimension here is close to 1. g2: Iterations on ± g3: Iterations on ± k -cycle, where k >
2. Julia setscan be very ”interesting”. The form of Julia set even more closely resemblesthe form of the tree. ”Fractality” is great and Hausdorff dimension isgreater, than 1 .
5. Good examples see in Figures 2, 3 and 6. g4:
Iterations on ± LANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIA SETS 5 ”Special” types.s1:
Iterations of 1 (for example) converge to an attracting point and iterationsof − k -cycle, k > s2: Iterations of 1 converge to one attracting k -cycle, k >
1, and iterations of − l -cycle, l > s3: Iterations of 1 (for example) converge to an attracting point and iterationsof − Remark . In what follows we will give several most interesting examples of Juliasets of SZ-polynomials. 4.
Trees with five edges
In this section for each 5-edge tree T we will compute its SZ-polynomial p T andfind characteristics of J ( p T ). The passport of a tree T is the list of degrees of whitevertices (in non increasing order) and the list of degrees of black vertices (also innon increasing order). We will always assume that ”white” list is lexicographicallynot less, than ”black” list.Estimations of Hausdorff dimension we will write in order: the box counting esti-mation, the packing estimation and the JP-algorithm estimation. If some methodis inapplicable, then we will put ”?” in the corresponding position.(1) Passport h , | , , , i . r rr❝ r ❝ T: ⇒ p T = (3 z + 1) (3 z − . Polynomial p T has an attracting 24-cycle. Iterations of ± p T is of g3-type. The set J ( p T ) is very similar to the set in Figure2. Hausdorff dimension: ≈ . ≈ .
32, ?.(2) Passport h , | , , , i . rr ❝ r ❝ r (cid:0)(cid:0)❅❅ T: ⇒ p T = − ( z + 2) ( z −
54 + 1 . Polynomial p T has an attracting 2-cycle:0 . → . → . − p T is of g2-type. Hausdorffdimension: ≈ . ≈ .
19, ?.(3) Passport h , , | , , i . rr ❝ r ❝❝ (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ T: ⇒ p T = − z + 10 z − z . Iterations of +1 and − p T is of g4-type. Julia setis totally disconnected. Hausdorff dimension: ?, ≈ . ≈ . YURY KOCHETKOV (4) Passport h , , | , , i . r ❝ rr ❝❝ (cid:0)(cid:0)❅❅ T: ⇒ p T = (2 z + 1) (2 z − z + 18)432 + 1 . Iterations of +1 and − p T is of g4. Hausdorffdimension: ?, ≈ . ≈ . h , , | , , i . r ❝ r ❝ r ❝ T: ⇒ p T = z − z + 5 z . The polynomial p T has two attracting 4-cycles:0 . → . → . → . → . − . → − . → − . → − . → − . − p T is of s2-type. Hausdorff dimension: ≈ . ≈ .
50, ?.5.
Trees with six edges
Only one non-symmetric 6 edge tree generates a connected Julia set:T: r rr❝ r ❝ r ⇒ p T = − z + 6 z + 4 z − z − z + 48 . The polynomial p T has a superattracting 2-cycle: 1 ↔ − p ′ T (1) = p ′ T ( −
1) = 0,i.e. p T is of g2-type. Hausdorff dimension: ≈ . ≈ .
15, ?. Julia set is similar toJulia set in Figure 4. 6.
Trees with seven edges
Here we have many trees that generate connected Julia sets. For such tree T wewill present behavior of iterations of ±
1, characteristics of Julia set J ( p T ) and thepicture of this set in interesting cases. In the picture of Julia set points that quitefast come into attracting domain of infinity (or into attracting domain of attractingpoint or a cycle) are white, points that come there more slowly are yellow, evenmore slowly are green, then light red, then deep red.We will use the following notations: • ± → ∞ means that iterations of 1 and − • ”p” means that SZ-polynomial has an attracting point; • ” c ( k )” means that SZ-polynomial has an attracting k -cycle; • ”1 → c (2) , − → c (3)” means that iterations of 1 converge to attracting2-cycle and iterations of − LANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIA SETS 7 h , | , , , , , i . r rr rr❝ r ❝ ✡✡✡❏❏❏ ❏❏❏ ✡✡✡ ⇒ ± → c (4); dim: 1 . , . , . p T is of g3-type. Julia set here is similar to Julia set in Figure 2.2) h , | , , , , , i . rr ❝ rr r ❝ r ⇒ ✑✑✑◗◗◗ ❇❇❇ ✂✂✂ ± → c (2); dim: 1 . , . , . p T is of g2-type. Julia set here is similar to Julia set in Figure 4.3) h , , | , , , , i . rr ❝ rr ❝❝ r ✑✑✑◗◗◗ ✂✂✂❇❇❇ ⇒ ± → p ; dim: 1 . , . , . p T is of g1-type. Convergence rate is quite good: | p ′ T ( p ) | ≈ . h , | , , , , , i . r r❝r r ❝ rr ❅❅(cid:0)(cid:0) ⇒ ± → p ; dim: 1 . , . , ?.Polynomial p T is of g1-type. Convergence rate is weak: | p ′ T ( p ) | ≈ . h , , | , , , , i . r ❝ r ❝rr r ❝ ⇒ ± → p ; dim: 1 . , . , . p T is of g1-type.6) h , , , | , , , i . ❝ r ❝rr r ❝❝ (cid:0)(cid:0)❅❅ ⇒ ± → c (4) ; dim: 1 . , . , . YURY KOCHETKOV
Polynomial p T is of g3-type. Here we have a high rate of convergence to theattracting cycle: the product of derivatives in cycle points is around 10 − . J ( p T ) : Figure 37) h , , , | , , , i . r ❝r❝r❝ r ❝ ⇒ ± → p ; dim: 1 . , . , ?.The polynomial p T is of g1-type.8) h , , | , , , , i . rr ❝ r ❝❝ rr (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ ⇒ ± → c (2) ; dim: 1 . , . , ?.The polynomial p T is of g2-type. Here we have a medium rate of convergence tothe attracting cycle: the product of derivatives in cycle points is around 0 . J ( p T ) : Figure 4 LANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIA SETS 9 h , , | , , , , i . r ❝ rr ❝❝ rr (cid:0)(cid:0)(cid:0)❅❅❅ ❅❅(cid:0)(cid:0) ⇒ ± → ∞ ; dim: ? , . , ?.The polynomial p T is of g4-type.10) h , , | , , , , i . rr ❝ r ❝ r ❝ r (cid:0)(cid:0)❅❅ ⇒ − → c (2) ; 1 → c (4) ; dim: 1 . , . , ?.The polynomial p T is of s2-type.11) h , , , | , , , i . ❝ rr ❝ r ❝❝ r (cid:0)(cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ ❅❅ ⇒ ± → p ; dim: 1 . , . , . p T is of g1-type.12) h , , , , | , , , i . r ❝ r ❝ rr ❝❝ (cid:0)(cid:0)❅❅ ⇒ − → p ; 1 → ∞ ; dim: 1 . , . , ?.The polynomial p T is of s3-type. J ( p T ) : Figure 5 Trees with eight edges
There are five trees whose SZ-polynomials have an attracting point, i.e. are ofg1-type. r r r r r rrr r (cid:0)(cid:0)❅❅ r r r r r rrr r (cid:0)(cid:0)❅❅ r r rrr rr rr ✟✟✟✟❍❍❍❍ r r rrr r rr r ✟✟◗◗◗ r r rrr r rr r ❍❍✑✑✑
There are three trees whose SZ-polynomials has an attracting 2-cycle, i.e. are ofg2-type. r rr r rr r r r ✡✡✡✡❏❏❏❏ rr r rr r r rr ✑✑◗◗✡✡❏❏ (cid:0)❅ r rr r rr rr r ✡✡✡✡✡❏❏❏❏❏
Next three cases are more interesting.1. r r r r rrr rr (cid:0)(cid:0)❅❅ ⇒ − → c (?) , → p. Polynomial p T is of s1-type.2. r r r r r r rrr ⇒ ± → c (7)Polynomial p T is of g3-type and its Julia set is visually interesting. J ( p T ) : Figure 6 LANE TREES, SHABAT-ZAPPONI POLYNOMIALS AND JULIA SETS 11 rr r r r rr r r ❅❅(cid:0)(cid:0) (cid:0)(cid:0)❅❅❅❅❅ ⇒ ± → c (7)It is an interesting example of g3-type polynomial p T that is not defined over R ,i.e. where T is not mirror symmetric.8. Some results about trees with big number of edges
If the passport is relatively simple, then SZ-polynomials can be computed for treeswith big number of edges. Here are some examples.
Example 8.1.
Passport h n, | , , . . . , i . If n >
7, then Julia set is totally dis-connected. Otherwise: n = 3 : ± → c (10); n = 4 : ± → c (24); n = 5 : ± → ∞ ; n = 6 : ± → c (4) . Passport h n, | , , . . . , i . If n >
13, then Julia set is totally disconnected. Other-wise: n = 3 : ± → c (2); n = 4 : ± → c (2); n = 5 : ± → c (2); n = 6 : ± → c (2); n = 7 : ± → c (4); n = 8 : ± → c (16); n = 9 : ± → ∞ ; n = 10 : ± → c (3); n = 11 : ± → ∞ ; n = 12 : ± → c (5) . Passport h n, | , , . . . , i . If 4 n
10 then ± → c (2). If n >
19, then Julia setis totally disconnected. Otherwise: n = 11 : ± → c (4); n = 12 : ± → c (4); n = 13 : ± → c (3); n = 14 : ± → c (5); n = 15 : ± → ∞ ; n = 16 : ± → c (3); n = 17 : ± → ∞ ; n = 18 : ± → c (6) . Passport h , , | , , , . . . , i .1) r rrrrrrrrrrrrrrr ± → ∞ r rrrrrrrrrrrrr rr ± → ∞ r rrrrrrrrrrrrrrr ± → ∞ r rrrrr rrrrrrrrrr ± → ∞ r rrrrrrrrrrrrrrr ± → p r rrrrrrrrrrrrrrr ± → p This example demonstrates that when we consider a set of trees with the samepassport, then almost all trees in this set have totally disconnected Julia sets, butfor ”nearly symmetric” trees this set is connected.9.
Conclusion
Further work in this field is related to the following problems.
Problem 1.
Prove that SZ-polynomials exist for all non-symmetric trees, or findan example of non-symmetric tree, for which SZ-polynomial does not exist.
Problem 2.
When SZ-polynomial p T of a tree T has an attracting cycle c ( k ), k > Problem 3.
Construct an analogue of SZ-polynomial for genus zero maps andstudy Julia sets for them.
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