How to check if one-dimensional solenoid in the sense of Williams can be realized as hyperbolic attractor of surface diffeomorphism Текст научной статьи по специальности «Математика»

MSC 2010: 37B05

How to check if one-dimensional solenoid in the sense of Williams can be realized as hyperbolic attractor of surface diffeomorphism

A. Yu. Zhirov

Moscow Aviation Institute (National Research University) Moscow 125993. E-mail: alexei_zhirov@mail.ru

Abstract. In the paper, an algorithm is presented that allows for a given one-dimensional solenoid in the sense of Williams to answer the question: does a surface diffeomorphism with a one-dimensional hyperbolic attractor exist such that the restriction of the diffeomorphism to this attractor is topologically conjugate to the shift automorphism of the given solenoid. Keywords: solenoid, hyperbolic attractor, branched manifold.

1. Introduction

In 1967 R. Williams [1] proposed the concept of generalized solenoid for modeling the dynamics generated by a diffeomorphism of a smooth manifold with a hyperbolic expanding attractor. The initial results of his theory are the following theorems. The conditions (S1-S4) specified in them will be formulated below.

Theorem 1 (Williams [2]). Let M be a smooth manifold, U С M be an open set, f : U ^ M be a smooth embedding, Л С U be a one-dimensional hyperbolic attractor for f. Then there exsist a branched 1-manifold K and a smooth map ф : K ^ K satisfying the conditions (S1-S4) such that the solenoid defined by the pair (K, ф) is homeomorphic Л, and its shift automorphism is topologically conjugate to the restriction f \л.

Theorem 2 (Williams [2]). Let K be a branched 1-manifold and ф : K ^ K be a smooth 'map satisfying the conditions (S1-S4). Then there exist a smooth manifold M, an open set U С M, a smooth embedding f : U ^ M, and a one-dimensional hyperbolic attractor Л С U for f such that the solenoid defined by the pair (K, ф) is homeomorphic to Л, and its shift automorphism is topologically conjugate to f \л.

For brevity, the statement of the theorem 2 will be said that the given solenoid can be realizable as an attractor on the manifold M.

Question. Under what conditions additional to those of the theorem 2 it can be confirmed that dimM = 2?

In the proposed paper, an algorithm will be presented with which the following questions can be answered for given solenoid.

© A. YU. ZHIROV

1. Whether the given solenoid is realizable as an attractor on an orientable surface and, if so, what is the minimal genus of a such surface? In this case the algorithm defines whether or not the embedding f generating the attractor preserves the orientation of the surface.

If the answer to the first question is negative, then

2. Whether the given solenoid is realizable as an attractor on an non-orientable surface and, if so, what is the minimal genus of a such surface?

2. Definitions and combinatorial description of solenoid

Let K be finite graph (one-dimensional CW-complex) and a its edge (open 1-cell), semi-edge is component of the set a \ {x}, where x E a. Two semi-edges are said to be equivalent if their intersection is semi-edge also. The equivalence class of semi-edges contained in a given edge is called its end. The end e of any edge is called to be incident to the vertex O if it is incident to semi-edge, which is a representative of the given end. The number of ends incident to a given vertex is called its valence.

Definition 1. Graph K is called branched (one-dimensional) manifold if the following conditions hold

(BM1) There are no vertexes of valency 1.

(BM2) For each vertex the partition of the set of ends of incident edges onto two nonempty disjunct subsets is fixed.

The ends belonging to one of these subsets will be called outgoing from a given vertex, and the ends belonging to the other one will be called incoming into it. Vertices of the graph will be called branch points of branched manifold K.

The smoothness of branched manifold will be understood in the sense of the following definition of the smoothness of the arcs lying in it. By an arc in K we mean an image by the action of a continuous mapping 7 of a segment of a real line into K, which is a local homeomorphism onto image.

Definition 2. The arc 7 C K is called to be smooth if for each t0 such that 7(t0) is branch point there exists e > 0 such that

1) arcs 7 I (to-s,t0) and 7 |(to,to+£) are semi-edges belonging to ends of some edges incident to this vertex 7(t0);

2) these ends belong to distinct subsets onto which the set of all edges incident to 7(t0) is parted according to condition (BM2), i.e. one of this ends is incoming while the other is outcoming.

A continuous map of a branched manifold K into a branched manifold K' (into a smooth manifold M) is called smooth if the image of every smooth arc in K is a smooth arc in K' in the sense of previous definition (smooth arc in M in the usual sense).

Definition 3. Let K be branched manifold and y : K — K be smooth map such that the following conditions hold

(51) the set of all branch points of K is invariant under y;

(52) the non-wandering set of y coincides with K;

(53) y is expanding in some metric;

(54) for each branch point there exists a neighborhood U and an integer m ^ 0 such that ymU is a smooth arc.

Solenoid in the sense of Williams is the limit of the inverse spectrum

K K K and the homeomorphism yc : Kc — Kc defined by

ycn : (X1,X2,X3, . ..) — (yXi,Xi,X2 .. .).

is called its shift automorphism.

Combinatorial description of the branched manifold 1) Let n be the number of edges of K. We enumerate in some way their ends by subscripts from the segment 1, 2n of the natural series and define the map a : 1, 2n — 1, 2n, assuming that j = a(i) if i,j are the numbers of the ends of the same edge. It is clear that a is a permutation on the set 1, 2n, decomposing into a product of cycles of period 2.

2) Let m be the number of vertices of the graph K. We enumerate them in some way by indices from the segment 1,m of the natural series and denote by t+ and t-, respectively, the number of outgoing and the number of incoming ends of the edges for the vertex number k. Denote by T = (t+,..., tt-,..., tm) the sequence of 2m numbers thus defined.

Thus the branched manifold defines a pair (a, t). Conversely, each pair of (a, t), where a is a permutation of the set 1, 2n, all cycles of which have a period 2, and T = (t+,..., t+m; t-,..., tm) is a sequence of 2m natural numbers with the condition Y^k=1(tt +1-) = 2n defines a branched 1-manifold. Therefore, we call the pair (a, T) associated with the branched manifold in the manner described above configuration of the branched 1-manifold. Of course, the configuration is ambiguous: it depends on the choice of the numbering of the ends of the edges, the numbering of the vertices, and which of the subsets of the ends in (BM2) are declared incoming and which are outgoing. Nevertheless, it is clear that knowing one of the configurations of a given branched manifold, one can write out all the others using a formal algorithm.

Combinatorial description of the map of branched manifold defining the solenoid. We enumerate the edges of the graph K with the index i E 1,n, endow

each of them with orientation, and denote the oriented edge number i by ai. Through ai we denote geometrically the same edge with opposite orientation. Let ф : K ^ K be a smooth map satisfying condition (S1). Then each edge ai is partitioned by preimages of branch points into arcs such that the restriction of ф to each of them is one to one map onto some edge that can be overpassed in either a positive or negative direction. Therefore, with each edge ai the word wi = ф(а^) in the alphabet A = {a1,...,an; a1,...,an} can be associated. From the smoothness condition it follows that this word is irreducible, i.e. does not contain subwords of the form aiai and aiai. Thus, with the map ф an ordered set of words w = (wi,... wn) be associated.

Conversely, a set of irreducible words w = (wi,...wn) defines a unique, up to homotopy, map ф of the graph K continuous on each edge and which is a local homeomorphism onto image in a neighborhood of each point that is not a branch point.

Now we establish the conditions under which this map is continuous at the branch points and satisfies the conditions (S1-S4) of the definition of solenoid.

Let's start with agreeing on the numbering of the vertices of the graph and its edges.

1. We number the vertices Ok with the index к E l,m.

2. Enumerate the ends of the edges, satisfying the condition: the outgoing ends for the vertex Oi get numbers from l to t+, outgoing ends for O2 — from t+ + l to t+ +t+ end so on; incoming ends for the vertex Oi get numbers from t + l to t +t-, incoming numbers ends to O2 — from t + t- + l to t + t- + t- and so on, where t = Y1 m=11+.

3. We enumerate the edges ai with the index i E l,n lexicographically with respect to the numbering of their ends: the smaller is the minimal number of two ends of the edge, the smaller is the number of the edge.

4. Orient the edges considering the positive direction from the end with the smaller number to the end with the larger one.

The end of the edge ai, which has a smaller number in the numbering of the ends (2), will be called first end, and the other end call the second one. These ends will be denoted by ai and ai respectively. (The fact that the same symbol denotes different things will not lead to misunderstandings, but it will be convenient.) The number of the first end of the edge ai in the "continuous" numbering of the ends will be denoted beta-(i), and the number the second denote (3+(i). The number of the edge that has one of its ends the end number j in continuous numbering will be denoted a(j). Thus, we obtain three maps of the sets of indices a : l, 2n ^ l,n; в-,в+ : l,n ^ l, 2n. It is easy to see that these functions are determined by the conditions

a(l) = l, ,a(i) = a(a(i)), a(i) < a(j) ^ min{i,a(i)} < min{j,a(j)};

в-j) := min {i : a(i) = j}; (5+(j) := max{i : a(i) = j};

The passage at the vertex Ok will be called an ordered pair of characters (two-letter word) in the alphabet A = {ai,ai : i E l,n} denoting ends of edges in this vertex, such that one of them belongs to the upper partition and the other to the lower one.

We will call pass £i£j image (where li is ai or ai, and is aj or aj) to be two-letter word, the first letter of which is the last letter of the word y(li) and the second is the first letter of y(lj).

Then the condition of continuity of the map y be

(SCO) Each two-letter subword of each word wi = y(ai) is a passage at some vertex. And the condition of smoothness of the map y be

(SC1) For each vertex, all its passes images are the passages at the same vertex.

Thus, calculating the images of passes, we can check the continuity condition y, and at the same time to find the map of the vertices w defined in the first part of the following

Definition 4. 1. The 'map of the vertices w is such permutation that

y(Ok) = Om(k).

2. The intersection matrix G = (gij) of the map y is defined by

9ij := #{ai,ai E Wj},

where # denotes the number of elements in finite set.

3. The intersection vector g = (gi) of the map y is defined by

gi :=Y1 gij.

j

4. The sequence E = {e]^,..., em} of signs £k = ±, depending of the way by which the map y turns the directions of passages on the vertex Ok into directions of passages

in °ro(k):

= j +, if "ingoing — outgoing" — "ingoing — outgoin"; £k \ —, if "ingoing — outgoin" — "outgoing — ingoing".

Lemma.

1. Conditions (S2) and (S3) together are equivalent to

(SC2) The intersection matrix G is primitive i.e. some power of it is a positive matrix.

2. Condition (S4) is equivalent to

(SC3) for each vertex, some iteration of all its passes is the same pass, up to the reverse.

Proof. According to the Perron-Frobenius theorem primitiveness and integerness of the matrix G imply that it has eigenvalue which is greater than one (Perron eigenvalue), which corresponds to positive eigenvector (Perron eigenvector). Therefore, it is possible to define on K a Lebesgue measure such that the length of each edge be equal to the

corresponding component of the left Perron eigenvector, and the map will be a uniform expanding with a coefficient equal to Perron eigenvalue. Since p is expanding and the matrix G is primitive, it does not have open invariant sets and the periodic points of p are dense in K. So primitiveness of the matrix implies conditions (S2,S3). The converse, as well as the second statement of the lemma, is obvious. □

All the above means that each one-dimensional solenoid define the set of data (a, T; w) by which it is possible to find the matrix G and the permutation w. This makes the following definition natural.

Definition 5. The collection of data (a, T; w), where a is permutation on 1,n with all cycles of the length 2; T = (t+,..., tt-,..., tm) is the sequence of 2m positive integers (m ^ 1) with the sum equal to 2n; w is the ordered set of irreducible words in alphabet A = {ai,ai : i E 1,n} is called solenoidal code if the condition (SC0-SC3) holds.

Of course, the code is determined by the solenoid ambiguously because of the following elements of arbitrariness

1. The choice of the numbering of vertexes.

2. The choice of which of the two possible subsets of the set of the ends of the edges incident to each vertex is considered to be the outgoing (ingoing).

3. The numbering of each of the sets of outgoing and ingoing edges of the edges.

In what follows, considering the solenoid given by any of its codes, we will save the

first two choices while we will have to consider all the codes obtaining while changing the choice of edge ends.

This means that the sequence will not change while the a permutation changes. The latter implies a change in the sequence of words w occurring due to a corresponding change in the numbering of the edges of the graph and a possible change in their orientations. The change in the numbering of the ends of the edges is determined by the permutation t : 1, 2n which leaves invariant each of the subsets of the set 1, 2n of indices of the form _

t— + 1,t+ (k e 1m, t+ = 0)

and

m

t +1— + 1,t +1- (k e 1m, t- = o,t = J21+).

k=1

Definition 6. We will call the permutation т of 1, 2n which leaves invariant the above subsets to be admissible.

If the admissible permutation is given we can to calculate the changed code (aT, T; wT) (T is the same) as follows.

The permutation of a is transformed into a permutation of aT, the cycles of which are obtained from the cycles of a follows. Let (i,j) be a cycle a. Then it corresponds to the cycle (т(i), т(j)) of aT. Of course it means that aT = т-1 о a о т.

To calculate the sequence of words wT, we note that the letters of the words Wj correspond to the cycles a. Namely, if (i, a(i), where i < a(i), then this cycle determines some letter aj. We will write aj ~ (i,a(i)), and the index of this letter can be defined as follows. We write the permutation a in the form of a product of such cycles, order the factors by the first index and enumerate them according to this ordering. Then j is the number of the cycle (i,a(i)) in this numbering. Keeping this ordering of cycles, replace each of them with the cycle (t(i), t(a(i))), keeping a sequence of characters in each cycle. The words wTk which define the map y a'k — wTk must be written in the alphabet A = {ai,a'i : i E 1, n} the letters of which denote the same edges of the graph, renumbered and, possibly, with a changed orientation. The letter aj,, denoting the same edge of the graph as the corresponding letter aj, is defined by the cycle (t(i),T(a(i))) and its number j' is the number of this cycle in their numbering in ascending smaller the indices of the permutation cycles of aT. The orientation of this edge coincides with its initial orientation i.e. aj, = aj, if t(i) < t(a(i)), and opposite to it otherwise, i.e. a j f a j .

Thus, to find the words wT, that define the map of y in new numbering of the ends is necessary in the correspondences aj — wj to make the appropriate replacement of letters a. and a-1 on aj, and a-1.

J J J J

Definition 7. We will say that the solenoid code obtained from given code by means of the described procedure using the admissible permutation t is t-equivalent to the initial one.

3. Band surfaces and codes of attractors

The solenoidal representation of a expanding one-dimensional hyperbolic attractor A of a diffeomorphism f : U — M in Williams Theorem 1 is obtained as follows. A tubular neighborhood N of the set A which is fibered onto (dim M — 1)-disks lying in the stable manifolds of the points of A is constructed. The partition of N onto these disks as leaves is semi-invariant with respect to f so that corresponding factorization of N defines branched manifold and its expanding map which define the solenoid.

In the case dim M = 2 the tubular neighborhood may be constructed in the form of band surface [3] i.e the surface n with boundary arranged as follows.

1. There is a finite collection of simple disjoint arcs (called base segments of n) I1,... ,Im of stable manifolds of some periodic points of f each of them containing corresponding periodic point. This collection is semi-invariant i.e. f (Ik) C Iw(k), where w is some permutation on 1,m.

2. There are two partitions D+ (upper partition) and D- (lower partition) of each base segment Ik onto t+ and t- segments (t+,t- ^ 1). Along some pairs A, A' eD := Uk D+ U D- of these segments a band n^A' is glued to the base I := (Jk Ik. The band surface is union of this bands. What is more, if x E A if nA,A' then the connected component of unstable manifold intersection Wu(x) f nA,A' containing x is the arc with endpoints on segments A, A' (called ends of the band) and the connected component

of stable manifold intersection Ws(x) П Пдд, containing x is the arc with endpoints on two other opposite edges of Пдд/ (called margins of band). The collection of all such arcs of stable manifolds defines partition of П onto 1-discs which is semi-invariant and define via factorization the branched 1-manifold K and its map ф satisfying to the conditions of definition of a solenoid.

We will also need a combinatorial description of the map f |n introduced in [3] to be a collection of data (a, T, o; w, Es, Eu, g, C) called the code of the attractor Л with respect to band surface П. These data formally are

1. a is permutation on 1,n (n the number of bands) all whose cycles are of period 2;

2. T = (t+,..., t+m; t-, tm) is a sequence of 2m natural numbers (m is the number of base segments);

3. o = [ol,..., on} — sequence of signs ±;

4. w — permutations on 1, m;

5. Es = [el,..., esn}, Eu = [eU,..., еП} — sequences of signs ±;

6. g — integer n-vector of natural entries;

7. C = [ll,... ,ln} finite integer sequence.

Definition 8. The data set listed above is called a formal code.

To define the geometric meaning of these data let us we introduce the following agreements.

Let us fix a positive direction (say "from the left to right") on each base segment; positive transverse direction (say "from bottom to top") to each of them. In the case of orientable surface suppose that each pair of these direction define the same orientation of the surface.

Now we assign the numbers to the ends A E D of the bands so that for Aj G DK and Aj/ G DK, (к, К = ±) i < i' if and only if either к = +, К = — or к = К and j < j' or к = к', j = j' and Aj lies on Ij to the left of A^.

So the permutation a be well defined and it means that the bands are glued to the base segments along the pairs of ends of the form (Aj, ACT(j)). The bands are numbered in such a way that the band nj is glued to segments Ap_j), Ae+(j) (fi-(j) < /3+(j),

a(e-(j)) = a(e+(j))= j).

The numbers t+,tk of the sequence T denote the numbers of ends of bands which are glued to the base segment Ik to the top and bottom, respectively.

The signs oj = — of the sequence o means that the band nj is twisted when gluing and oj = + in other case. In the case of an orientable surface, there are no twisted tapes, while for non-orientable ones they should be.

The triple (a, T, o) is called the configuration of the band surface.

The remaining data of the code characterize the action of the map fn on the base segments and bands.

Permutation w shows that fnIk = I^(k).

Signs £sk are " + "ore " — "depending of whether fn move positive direction of Ik into positive ore negative direction of I^(k). Signs £U means the same with respect transverse directions.

Component gi of the vector g is the common number of component of intersection of the band ni with images of other bands.

Element lk of the sequence L is she number of components of intersection of images of all bands with that component of Ik \ f (Im(-i)), which is to the left of f (I^-i) (lk = 0 if left end of f (I^-i) coincides with that one of Ik).

In [3], it is shown (Lemma 2.4.1) that the code of the attractor satisfies some conditions that need to be formulated here. This requires the following notation. By formal configuration (a, T, o) define integral vectors V+ (j E 0, t) and V- (j E t,t + u, t = £ t+, u = £ t-)

V+ = V- := 0; V± := V- + ea(j). Here ei is the vector with all entries 0 except for i-th which is 1.

v+ := V+ — V+k-i; v- := V+m+k — V+m+fc-i, k E 1,m

Here Tk = Y1 kj=11+ for k ^ m and Tm+k = t + Yjj=11-.

In these terms, the conditions mentioned are (FC1) <V^(k) + V^k), g) ^ t+ + t-; (FC2) <V^(k) — V^(k), g) = £U(t+ — t-); (FC3) 0 ^ lMk) ^ 1 (<V-(k) + V^(k), g) — (t+ + t-)). Here <•, •) denotes the usual scalar product.

Definition 9. The formal code is said to be admissible if it satisfies to conditions (FC1-FC3).

In [3] the conditions (FC1-FC2) immediately included in the definition of a formal code.

We note that condition (FC3) is especially important now, since it implies the finiteness of the set of formal codes with given n and m, and an estimate of their number can be easily obtained.

Solenoidal representation of attractor A is obtained from band surface n and the map f |n as follows. Branch points (vertexes) of branch manifold K are images under factorization of base intervals Ik and them are numbered by the same indexes. So, the permutation w is the same as permutation of branch points. Partitions of ends edges of K onto ingoing and outgoing ones corresponds to partitions of segments Ik onto upper and lower ends of bands D+ and D-. Edges ai of K are factors of bands n

and are enumerated y the same indexes. Orientation of the edge aj corresponds to the "vertical"direction on nj i.e. direction from its end Ap_(j) to A^+(j).

The Theorem 3.1.1 [3] implies that each solenoidal representation of an attractor is obtained by the method described above.

4. Algorithm for checking the realizability of a solenoid on the surface

The data a, T, w which give combinatorial description of solenoid enables to find a part of the data composing the code of the attractor: a, T, w, g, Eu = E. Hence, if the solenoid is given, it is possible to enlarge these data to obtain the formal code. This can be done in various ways of specifying missing data o, Es, C. What is more, in the source data we can replace a and w by performing a permissible permutation т of the indices. In any case, the number of opportunities to do all this is finite and can be estimated through initial data. Thus, it is possible to consider all the various additions of the solenoidal code to the formal code.

In [3] an algorithm for checking if the formal code is realizable by some attractor via some band representation of it is described. It is based on the some ancillary algorithms (subroutines in terms of computer sciences) i.e. so called Algorithms A, B and D. We will not give a description of these algorithms here because it requires too much space, but it is necessary to specify their input and output data along with a brief comments.

Algorithm A.

Input. Formal (n, m)-configuration (a, T, o).

Output. 1. The sequence B = [b0, bl,...} of nonnegative integers such that almost all (but not all) of them are equal to zero.

2. The finite set of nonempty words R = [p1,..., pb} in alphabet A = [aj,aj : i G

In}.

Comment. In the case when the configuration on input is indeed that one of some band surface of the attractor we have b0 = 0 (it is essential for our goals), while other elements of B define so called boundary type of the attractor (see [3] for definition, we need not for it now). The set of words R defines some co-representation (al,..., anlpl,..., pb) of fundamental group of the band surface.

Algorithm B.

Input. Formal (n, m)-code (a, T, o; w, Es, Eu, g, C)

Output.

1. Formal configuration (a, T, o).

2. Ordered set W = [wl,... ,wn} in alphabet A = [aj,aj : i G 1,n}.

3. Sequence C = [ll,..., lm} of non-negative integers.

Comment. If the formal code on input is indeed a code of some attractor then the data on output (a, T, o) and C are the same as on input, while the sequence of words WW is that one which define the map ф : K -л K which defines solenoid. In any case WW

defines in the same way as w some integer matrix G and the corresponding vector g. We will need of them to check the condition of realizability.

Algorithm D.

Input. Formal (n, m)-code (a, T, o; w, Es, Eu, g, L); integer N > 1. Output. Formal (n, m)-code (a, T, o; w(N), Es(N), Eu(N), g(N), L(N))

Comment. If the formal code on input is indeed a code of some attractor of f then the code on output is the code of the same attractor considered as that one with respect to N-th iteration of f.

This algorithm of checking realizability of solenoid as an attractor on surface is based on the following criteria.

Theorem 3 ([3], Theorems 3.8.1, 3.8.2). The following conditions are necessary and sufficient for the realizability of the formal code (a, t, o; w, Es, Eu, g, L).

(RC0) The number b0 calculated by algorithm A is zero and the formal configuration (a, T, o) calculated by algorithm B coincides with the initial one.

(RC0) The vector composed of the sum of rows of the matrix G calculated by Algorithm, B coincides with g.

(RC2) None of the words in the set [<g(p) : p E R} is cyclically trivial. Here фт is the 'map ai M Wj,.

(RC3) The matrix G is primitive.

(RC4) For minimal N E N such that w(N) = id, Es(N) = Eu(N) = + and for each к E 1,m the following conditions hold

(VT+k-1+i - VT+k-1 )g(N) = if] + г for all г E 1,t+ - 1 (ytm+k-1+i - утт+к-1 )g(N) = i(N) + г for all г e 1,t- - 1

Now we are ready to formulate the desired algorithm. Main algorithm

Input. An abstract 1-solenoid defined by a branched manifold with m branch points and n edges given by its solenoidal code (a, T; w). (The code must satisfy the conditions (SC0-SC3)).

Move 0. By the sequence of words w define the permutation w and the sequence of signs E (Definition 4).

Move 1. Enumerate all admissible permutations of т and for each (including the identity) calculate the transformed code (aT, T; wT).

Move 2. For each of them (we omit the index т below) define a vector g by a sequence of words w and write down all formal codes (a, t, o; w, Es, Eu, g, L) with o = (+,..., +), Eu = E, Es = Eu or Es = -Eu and L satisfying to the condition (FC3).

Move 3. For each of these formal codes check the conditions of Theorem C and, if they are not satisfied for any of them, to conclude that this solenoid is not realizable on any oriented .surface. Go to move 6.

Move 4. For each code that satisfies the conditions of Theorem C, check equality W = w, where W is the sequence of words which is calculated for using Algorithm B and w is that one from initial solenoidal code. If it is not satisfied for any of them, make the same conclusion.

Move 5. If we find a code that satisfies all these conditions then the solenoid is realizable on orientable surface. And calculating for these codes (using the Algorithm A) a set of words R we define via corresponding co-representation of fundamental group topological types of these surfaces. So we obtain minimal genus of such surface. What is more, if Es = Eu then generating the attractor diffeomorphism preserves orientation and it reverses it in the case Es = —Eu. The end of the procedure of the Algorithm,.

Move 6. For each of transformed solenoidal codes calculated on the Move 1 write down all formal codes with all sequences o with at least one entry "—", Eu = E, all possible sequences Es and all sequences L satisfying to the condition (FC3).

Moves 7 and 8. For each of these codes check the same conditions as on the Moves 4 and 5. And if these are not valid for each of them than the solenoid is not realizable on any surface.

Move 9. If we find a code that satisfies all these conditions then the solenoid is realizable on non-orientable surface and we can define minimal genus of such one in the same way as in orientable case using Algorithm A. The end of the procedure of the Algorithm.

References

1. Williams, R. F. One-dimensional non-wandering sets // Topology. — 1967. — T.6, №4. — C. 473-487.

2. Williams, R. F. Expanding attractors // Publ. Math. Inst. Hautes Etud. Sci. — 1973. — T.43. — C. 161-203.

3. Zhirov, A. Yu. Combinatorics of one-dimensional hyperbolic attractors of diffeomorphisms of surfaces // Proceeding of the Steclov Insitute of Mathematics. — 2004. — T.244. — C. 132-200.

Получена 05.10.2018