Title Local inverses of shift maps along orbits of ... - Osaka University

TitleLocal inverses of shift maps along 

Author(s) Maksymenko, Sergiy 

Citation Osaka Journal of Mathematics. 48(2) 

Issue 2011-06 

Date 

Text Version 

publisher 

URL 

http://hdl.handle.net/11094/10155 

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Osaka University

416 S. MAKSYMENKO 

denote by Sh(F) its image ³(C½ (V ,Ê)) in C½ (M, M). Then the following inclusions 

hold true: 

1 Sh(F)E id (F)½¡¡¡E id (F) E id (F) 0 . 

The idea of replacing the time in a flow map with a function was extensively used 

e.g. in [8, 3, 31, 12, 27, 10] for reparametrizations of measure preserving flows and investigations 

of their mixing properties, see also [19, §1]. Smooth shifts functions were 

applied in [13, 15, 16, 14, 17] for study of homotopical properties of certain infinite 

dimensional groups of diffeomorphisms and their actions on spaces of smooth maps. 

Also in [19, 18] some applications to parameter rigidity of vector fields are given. 

Suppose F is linearizable at each z ¾ F, see e.g. [29, 30, 32, 2, 11]. In [13] 

the author in particular claimed that if in addition M is compact, then D id (F) 0 and 

E id (F) 0 are either contractible or homotopy equivalent to the circle S 1 when endowed 

with W½ -topologies. Unfortunately, it turned out that in such a generality this statement 

fails and it is necessary to put additional restrictions on the behavior of regular 

orbits of F. In fact it was shown that Sh(F)E id (F) 0 and the mistakes appeared in 

estimations of continuity of local inverses of ³, see [13, Definition 15, Theorem 17, 

Lemma 32] and also Remarks 7.4 and 8.1 for detailed discussion. 

Furthermore, the above description of D id (F) 0 was essentially used in another author’s 

paper [15] concerning calculations of the homotopy types of stabilizers and orbits 

of Morse functions on surfaces. 

The aim of the present paper is to repair incorrect formulations and proofs of [13] 

and using [14, 19, 17] extend the classes of vector fields for which the homotopy types 

of E id (F) r and D id (F) r can be described (Theorem 3.5). In particular it will be shown 

that the results of [13] used in [15] remain true (Theorem 3.7). 

1.1. Structure of the paper. For the convenience of the reader and due to the 

length of the paper we will now briefly describe its contents. 

In §2 we recall the notion of shift map and review (correct) results obtained in 

[13]. Also notice that if a flow F of a vector field F is not global, then we can find 

a smooth function Ï M (0,½) such that the flow F¼ 

generated by vector field 

F¼ F is global, e.g. [6, Corollary 2]. Moreover, F and F¼ have the same orbit 

structure, whence E(F)E(F¼ ) and D(F)D(F¼ ). Thus it could always be assumed 

that F is global. Nevertheless, in §10 and §11 we will consider restriction FW of 

F to open subsets W M and compare shift maps of F and FW . The latter vector 

field usually generates non-global flow, therefore in this paper it is chosen to work with 

local flows from the beginning. In particular, the domain of shift map ³ of F changes 

to certain open and convex subset func(F)C½ (M,Ê). 

In §3 we introduce a certain class of vector fields F(M) on M and formulate the principal 

result of the paper: for every F¾F(M) its shift map³ is a local homeomorphism of 

func(F) onto its image Sh(F) with respect to S½ -topologies (Theorem 3.5). It follows that

LOCAL INVERSES OF SHIFT MAPS 417 

Sh(F) is either contractible or homotopy equivalent to the circle, and so is the subspace of 

Sh(F) consisting of diffeomorphisms. On the other hand, in [18] it is considered a class of 

vector fields F¼ (M) which contains F(M) and such that for every F ¾ F¼ (M) the image 

of its shift map Sh(F) coincides with either E id (F) 1 or E id (F) 0 for each F ¾ F(M). Thus 

we obtain that E id (F) r (D id (F) r ) coincide with each other at least for all r 1 and these 

spaces are either contractible or have homotopy type of S 1 . As an application we prove 

Theorem 3.7 which extends [15, Theorem 1.3]. The rest of the paper is devoted to the 

proof of Theorem 3.5. 

The assumptions of Theorem 3.5 imply that ³ is locally injective, therefore in 

order to prove this theorem it suffices to show that ³ is open with respect to S½ topologies. 

In §4 we present a characterization of S r,s -openness of ³ for some r, s 

0, (Theorem 4.2). First we describe local inverses of ³V as certain crossed homomorphisms 

and then show that like for homomorphisms of groups S r,s -openness of the 

whole map ³ is equivalent to S s,r -continuity of its local inverse defined only on arbitrary 

small S s -neighbourhood of the identity map id M . 

Further in §5 we recall notorious examples of irrational flows on the torus and 

some modifications of them. It is shown that shift maps of these flows are not local 

homeomorphisms onto their images. This provides counterexamples to [13, Theorem 1] 

and illustrates certain properties prohibited by Theorem 3.5. 

In §6 we repair [13, Lemma 28] by giving sufficient conditions for S r,s -openness 

of ³ to be inherited by regular extensions of vector fields. 

§7 summarizes the formulas for local inverses of shift maps of linear flows obtained 

in [13] and “reduced” Hamiltonian flows of homogeneous polynomials in two 

variables obtained in [19, 14]. Estimations of continuity of these formulas for linear 

flows were based on incorrect “division lemma” [13, Lemma 32], see Remark 7.4. We 

will show how to avoid referring to this lemma. 

§8 provides a correct version of [13, Theorem 17], see Theorem 8.2. It reduces 

verification of openness of ³ to openness of a family ³V i of “local shift maps” corresponding 

to any locally finite cover V i of M e.g. by arbitrary small smooth closed 

disks. The essentially new additional object here is a finite subset ¿¼¿ which appears 

due to the construction (8.3) of S½ -open set N and by existence of singularities 

for which “local shift map” is S½,½ -open but not S r,d(r) -open for some function 

dÏÆÆ. Without finiteness of ¿¼ the shift map ³ is open if its image is endowed 

with the so-called very-strong topology, [9]. This effect appears of course only on noncompact 

manifolds. 

§9 splits the “global analytical problem” of verification of openness of local shift 

map ³V into the following two problems 

(i) openness of local shift map³W ,V corresponding to the restriction FW to arbitrary 

open neighbourhood W of V , and 

(ii) openness of the image Sh(F W , V ) in Sh(F, V ). 

Due to Theorem 8.2 the set V can also be taken arbitrary small. Hence to solve (i) 

it suffices to consider vector fields in Ên . Thus (i) is a “local analytical” problem. Its


Fig. 1. D-submanifolds V and W . 

solutions for certain vector fields are given in §7. 

§10 and §11 present sufficient conditions for resolving (ii) at regular and singular 

points respectively (Theorem 10.1 and Lemma 11.2). In both cases the assumptions on 

F are formulated in the terms of dynamical systems theory, and so the problem (ii) 

can be regarded as a “global topological” one. In the regular case these conditions are 

also necessary. Moreover, in the singular case they are relevant with the notion of an 

isolated block introduced by C. Conley and R. Easton in [4], see Lemma 11.5. 

Finally in §12 we prove Theorem 3.5. 

1.2. Preliminaries. Put Æ0Æ0 and Æ0Æ0,½. 

Let M be a smooth manifold of dimension m. Glue to M a collar M¢ [0, 1) by 

the identity map M¢ 0 M and denote the obtained manifold by M¼ . If Mû, 

then M¼M. Evidently, M¼ has a smooth structure in which it is diffeomorphic with 

the interior Int M. Moreover M is a closed subset of M¼ and M¼û. We will call 

M¼ a collaring of M, see Fig. 1. 

DEFINITION 1.3. 

A closed subset V M will be called a D-submanifold if there 

exists an m-dimensional submanifold V¼ M¼ possibly with boundary and such that 

V M V¼ and the intersection MV¼ is transversal, see Fig. 1. 

Denote by Int V the interior of V in M. Then Int V VÒV¼ . We will also say 

that V is a D-neighbourhood of every point z¾ Int V . 

Let V M be a D-submanifold. If V M û, then V is a manifold with 

boundary, otherwise, V is a manifold with corners in which MV¼ is set of corners. 

Evidently, M is a D-submanifold of itself, and each z¾ Int M has arbitrary small 

D-neighbourhoods each diffeomorphic to a closed m-disk. 

Let N be another smooth manifold, N¼ be its collaring, and fÏ V N be a map. 

We say that f is of class C r , r ¾Æ0, (embedding, immersion, etc.) if it extends to a 

C r 

map (embedding, immersion, etc.) U N¼ from some open neighbourhood U of 

V in M¼ into the collaring N¼ of N. 

M is C r 

Then it is well-known that f is C r if, and only if, the restriction fInt VÒ 

and all its derivatives have continuous limits when x tends to some point y¾V¼ , see 

e.g. [33, 28, 24, 5] for manifolds with boundary and e.g. [22, Proposition 2.1.10] for 

manifolds with corners.


Denote by C r (V , N), r ¾Æ0, the space of all C r maps V N. Then similarly 

to [7] this space can be endowed with weak W r - and strong S r -topologies. If V is 

compact then W r and S r coincide for all r¾Æ0. For a subset X C r (V , N) denote by 

(X ) r W (resp. (X )r S ) the set X with the induced Wr -topology (resp. S r -topology), r¾Æ0. 

DEFINITION 1.4. A homotopy HÏ V¢ I N will be called an r-homotopy, (r¾ 

Æ0), if for every t¾I the map H t H(¡, t)Ï V N is C r and all its partial derivatives 

in x ¾ V up to order r are continuous in (x, t). In other words, H is an r-homotopy 

if and only if it yields a continuous path I C r (V , N) from the standard topology of 

I to the W r -topology of C r (V , N). 

Hence for every f ¾ X its path connected component in (X ) r W 

consists of all g¾ 

X which are r-homotopic to f in X . 

Now let V 1 , V 2 be D-submanifolds of some smooth manifolds, N 1 , N 2 be two smooth 

manifolds, X C½ (V1 , N 1 ) and Y C½ (V2 , N 2 ) be two subsets, and FÏ X Y be 

a map. 


We say that F is W r,s -continuous (-open, etc.) for some r, s¾ 

(Y) s W . 

Æ0 if it is continuous (open, etc.) as a map FÏ (X ) r W 

 

Similarly we can define S r,s -continuous (-open, etc.) maps with respect to strong 

topologies. 

Notice that the statement that F is W½,½ -continuous at some x ¾ X means that 

for every s 0 and a W s -open neighbourhood V F(x)Y of F(x) there exist r 0 and 

a W r -neighbourhood U x X of x both depending on s and V F(x) such that F(U x ) 

V F(x) . But in general such a map can be not W r,s -continuous for any r, s, see e.g. [23, 

p. 93 Equation (2)], and [25]. 

2. Shift map 

M such that o x (0)x and 

is the maximal interval on which a map with the 

previous two properties can be defined. By standard theorems in ODE the following 

subset of M¢Ê 

dom(F) x¢ (a x , b x ), 

Let F be a vector field on M tangent to M. Then for every x ¾ M its orbit 

with respect to F is a unique mapping o xÏÊ(a x , b x ) 

(ddt)o x F(o x ), where (a x , b x )Ê 

x¾M 

is an open, connected neighbourhood of M¢ 0 in M¢Ê. Then the local flow of F is 

the following map 

FÏ M¢Êdom(F) M, F(x, t)o x (t).


Fig. 2. Domain dom(F) of the flow F of the vector field F(x) 

x 2 ddx. 

If M is compact, then dom(F) M¢Ê, e.g. [26]. 

Notice that if x¾M is either fixed or periodic for F, then x¢Êdom(F). 

EXAMPLE 2.1. Let F(x)x 2 ddx be a vector field on Ê1 . Then it is easy to 

see that it generates the following local flow F(x, t) x(1 xt). Hence F is defined 

on the subset dom(F)Ê2 bounded by the hyperbola xt 1, see Fig. 2. 

Let V M be either an open subset or a D-submanifold and i VÏ V M be 

the identity inclusion. Denote by E(F, V ) the subset of C½ (V , M) consisting of maps 

fÏ V M such that (i) f (oV )o for every orbit o of F, and (ii) f is an embedding 

at every singular point z¾VF. Let also D(F, V )E(F, V ) be the subset 

consisting of immersions V M. 

Let E id (F, V ) r and D id (F, V ) r be the path-components of i V in the spaces E(F, V ) 

and D(F, V ) respectively endowed with the corresponding topologies W r . 

Denote by func(F, V ) the subset of C½ (V ,Ê) consisting of functions « whose 

graph¼«(x,«(x))Ï x¾ V is contained in dom(F). Then we can define the following 

map 

³VÏ C½ (V ,Ê)func(F, V )C½ (V , M) 

by³V («)(x)F(x,«(x)). We will call³V the (local) shift map of F on V and denote 

its image in C½ (M, M) by Sh(F, V ). Put 

¼V «¾ func(F, V )Ï d«(F)(x) 1, x¾ V, 

where d«(F) is a Lie derivative of « along F. Since func(F, V ) is an S 0 -open and 

convex subset of C½ (V ,Ê) and the map 

L FÏ C½ (V ,Ê)C½ (V ,Ê), 

L F («)d«(F) 

is evidently linear and S 1,0 continuous, we see that¼V is convex and S1 -open inC½ (V ,Ê).


It also follows from [13, Lemma 20 and Corollary 21] that 

(2.1) ¼V ³ 

1 

V (D id(F, V ) 1 ), Sh(F, V )E(F, V ). 

Lemma 2.2 ([17]). 

³V (¼V 

The following inclusions hold true: 

)D id(F, V )½¡¡¡D id (F, V ) r ¡¡¡D 

Sh(F, V )E id (F, V )½¡¡¡E id (F, V ) r ¡¡¡E 

If Sh(F, V )E id (F, V ) r , then ³V (¼V )D id(F, V ) r . 

The set ker(³V ) ³ def 1 

V (i V ) will be called the kernel of ³V . 

id (F, V ) 0 , 

id (F, V ) 0 . 

Lemma 2.3. The following properties of shift map hold true: 

(1) ³V is W r,r - and S r,r -continuous for all r 0, [13, Lemma 2]. 

(2) d(F)0 for every ¾ker(³V ), [13, Lemma 7]. 

(3) Let«,¬¾ func(F, V ). Then, [13, Lemma 7], the following conditions are equivalent: 

(a) ³V («)³V (¬) 

(b) « ¬¾ func(F, V ) and ³V (« ¬)i V , i.e.« ¬¾ ker(³V ). 

(4) ³V is locally injective with respect to each of W r - or S r -topologies of func(F, V ) 

if and only if F V is nowhere dense in V , [13, Proposition 14]. 

(5) Suppose that V is connected and that VF is nowhere dense in V . Then, [13, 

Theorem 12 (2)], 

(a) either ker(³V )0 and thus ³V is injective. This case holds if V contains 

either a non-periodic point of F or a fixed point z¾VF such that the tangent 

linear flow T z F t on the tangent space T z M is the identity; 

(b) or ker(³V )nn¾ for some smooth Ï V (0,½). In this case V¢Ê 

dom(F), so func(F, V )C½ (V ,Ê). 

3. Main result 

Let M be a smooth manifold of dimension m. We will introduce a class F(M) 

of smooth vector fields F on M satisfying certain topological conditions on regular 

points and certain analytical conditions on singular points. The main result describes 

the homotopy types of the identity path components of E(F) and D(F). First we give 

some definitions. 

Recurrent points. Let F be a vector field on M and z¾M be a regular point 

of F, i.e. F(z)0. Then z is called recurrent if there exists a sequence t ii¾Æ of 

real numbers such that lim i½t i½ and lim i½ F(z, t i )z. In particular, every 

periodic point is recurrent.


First recurrence map. Let z be a periodic point of F, o z be its orbit, and B 

M be an open (m 1)-disk passing through z and transversal to o z . Then we can 

define a germ at z of the so-called first return (or Poincaré) map RÏ (B, z)(B, z) 

associating to every x¾B the first point R(x) at which the orbit o x of x first returns 

to B. It is well-known that R is a diffeomorphism at z, e.g. [26]. 

Reduced Hamiltonian vector fields. Let gÏÊ2 Ê be a homogeneous polynomial 

of degree p2, so we can write 

(3.1) gL l 1 

1 

¡¡¡ L l a 

a ¡ Q q 1 

1 

¡¡¡ Q q b 

b , 

1, L iL i¼ const for i i¼ , and Q jQ j¼ const for j j¼ . Denote 

where L i is a non-zero linear function, Q j is an irreducible overÊ(definite) quadratic 

form, l i , q j 

DL l 1 1 

1 

¡¡¡ L l a 1 

a ¡ Q q 1 1 

1 

¡¡¡ Q q b 1 

b 

. 

Then gL 1¡¡¡L a¡Q 1¡¡¡ Q b¡D and D is the greatest common divisor of the partial 

derivatives g¼x and g ¼y . The following vector field on Ê2 : 

F(x, y) g ¼y 

D 

 

x 

g ¼x 

D 

will be called the reduced Hamiltonian vector field of g. In particular, if g has no 

multiple factors, i.e. l i q j 1 for all i, j, then D 1 and F is the usual Hamiltonian 

vector field of g. 

Notice that dg(F) 0 and the coordinate functions of F are homogeneous 

polynomials of degree deg F a 2b 1 being relatively prime in the ring of polynomials 

Ê[x, y]. 

“Elementary” singularities. Let F be a vector field onÊk . It can be regarded as 

a map FÏÊkÊk . Define the following types (Z), (L), and (H) of such vector fields. 

Type (Z). F(x)0. 

Type (L). F(x)«(x)¡ Ax, where A is a non-zero (k¢ k)-matrix, and «ÏÊn 

(0,½) is a C½ strictly positive function. Let 1,,k be all the eigen values of 

A taken with their multiplicities. We can also assume that A has a real Jordan normal 

form. Then we will distinguish the following particular cases. 

(L1) ( j)0 for some j 1,, k; 

(L2) The real Jordan normal form has a block 

 

 

0 1 

0 1 

¡¡¡ ¡¡¡ 

0 1 

0 

 

n, 

y 

n 2Á


(L3) The real Jordan normal form has a block 

 

0 b 1 0 

b 0 0 1 

¡¡¡ ¡¡¡ ¡¡¡ 

0 b 

b 0 

1 0 

0 1 

0 b 

b 0 

 

 

2n, 

n 2, b0. 

In all other cases A is similar to the matrix: 

 

0 b 1 

b 1 0 

¡¡¡ 

0 b n 

b n 0 

0 

¡¡¡ 

0 

, n 1, b j 0, j 1,, n. 

 

Then we will also distinguish the following two cases: 

(L4) there exists 0 such that b j ¾ for all j; 

(L5) such a number as in (L4) does not exist. 

Type (H). F(x, y)«(x, y)¡( g¼yD, g¼xD) for some strictly positive 

tion«ÏÊ2 

C½ func- 

(0,½), and a homogeneous polynomial (3.1) such that a 2b 12, 

so F is not of type (L). Again we separate the following cases: 

(HE) a 0 and b2, i.e. g is a product of at least two distinct definite quadratic 

forms and has no linear multiples. In this case 0¾Ê2 is a degenerate global extreme 

of g; 

(HS) a 1 and a 2b3, so g has linear multiples. In this case 0¾Ê2 is a degenerate 

saddle critical point for g. 

REMARK 3.1. The types (L), (H), (HE), and (HS) coincide with the ones considered 

in [18]. 

Also notice that in the case (L4) almost all orbits of F are periodic, while in the 

case (L5) almost all orbits are non-periodic and recurrent. 

Regular extensions and products of vector fields. Let M, N be manifolds, 

GÏ M T M be a vector field on M, and F be a vector field on M¢N. Then F can 

be regarded as a map FÏ M¢N T M¢ T N. We say that F is a regular extension 

of G if 

F(x, y)(G(x), H(x, y)), 

(x, y)¾ M¢N,


for some smooth map HÏ M¢N T N, so G is the “first” coordinate function of 

F and does not depend on y, see e.g. [1, 32]. If H does not depend on x, so it is a 

vector field on N, then 

F(x, y)(G(x), H(y)) 

will be called the product of G and H. Moreover, if H 0, i.e. 

F(x, y)(G(y), 0), 

then F is said to be a trivial extension of G. 

In the case N Ên we will also say that F is a (regular or trivial) n-extension. 

Evidently, a regular (trivial) m-extension of a regular (trivial) n-extension is a regular 

(trivial) (m n)-extension, see e.g. [32, 13]. The following simple statement is left for 

the reader: 

Lemma 3.2. (a) A trivial extension is the same as a product with zero vector field; 

(b) a product of vector fields is a regular extension of each of them; 

(c) every non-zero linear vector field is a regular extension of a linear vector field 

defined by one of the following matrices: a, a b , (b0), or 

0 1 

 

b a 

0 0. 

DEFINITION 3.3. Let z¾F and (T) denotes one of the types (Z), (L), (L1), etc., 

defined above. We will say that the germ of F at z is of type (T)¼ 

if the germ of F 

at z is equivalent to a regular extension of some vector field of type (T). By (T) and 

(T)¼ 

we will denote the set of singular points of F or types (T) and (T)¼ 

respectively. 

Evidently, a singular point can belong to distinct types. 

We will say that a C½ vector field F on a C ½ manifold M 


belongs to class F(M) if it satisfies the following conditions: 

(a) F is tangent to M and F is nowhere dense in M; 

(b) every non-periodic regular point of F is non-recurrent; 

(c) for every periodic point z of F the germ at z of its first recurrence map RÏ (D, z) 

(D, z) is either periodic or the tangent map T z RÏ T z DT z D has eigen valuesuch that 

1; 

(d) for every z¾F the germ of F at z is either 

(£) a product of finitely many vector fields each of which is of type (L) or (H), 

or 

(££) belongs to one of the types (L1)¼ , (L2) 

¼ , (L3) 

¼ , or (H) 

¼ , but z is an isolated 

singular point of F. 

Thus if the germ of F at z is a regular extension of an (L4)-vector field, then F 

must in fact be a product of such a vector field with vector fields of types (L) or (H)


only. Also notice that singularities of type (L5) are not allowed at all since they have 

recurrent orbits. Moreover, we will present examples of vector fields with singularities 

of type (L5) for which the statement of our main result fails, see the end of §5. 

The following theorem summarizes the results obtained in [18] and in this paper. 

Theorem 3.5 (cf. [13, Theorem 1] and [18]). Let F¾F(M),³ be the shift map 

of F, and ¼«¾ func(F)Ï d«(F)(x) 1, x¾M. Then 

Sh(F)E id (F) 1 , ³(¼ )D id (F) 1 . 

If (L)¼(HS)¼ 

, then Sh(F)E id (F) 0 and ³(¼ )D id (F) 0 . 

Suppose that the closure Ò ((L1)¼(L2)¼ 

) is compact (this is always true for 

compact M). Then both maps 

³Ï func(F)E id (F) 1 , ³¼Ï¼ D id (F) 1 

are either homeomorphisms or -covering maps with respect to S½ -topologies. If M 

is compact, then the inclusion D id (F) 1 E id (F) 1 is a homotopy equivalence and both 

spaces are either contractible or homotopy equivalent to the circle. If F has at least 

one non-closed orbit, or a singular point at which the linear part (i.e. 1-jet) of F vanishes, 

then D id (F) 1 and E id (F) 1 are contractible. 

The first statement about the image of shift maps is established in [18] under more 

general assumptions on F. Therefore we will be proving the second part of Theorem 3.5, 

see §12. Notice that in comparison with [13, Theorem 1] two additional assumptions are 

added: “non-ergodicity” condition (like absence of recurrent orbits) and compactness of 

certain subset of singular points. On the other hand the classes of admissible singularities 

for F are extended due to results of [14, 19, 20]. 

3.6. Functions on surfaces. As an application of Theorem 3.5 we will now 

show that [15, Theorem 1.3] which used incorrect results of [13] remains true. Moreover, 

we extend the latter theorem to a large class of functions on surfaces with degenerate 

homogeneous singularities satisfying certain “secondary” non-degeneracy conditions. 

Let M be a compact surface, P be either the real line Ê or the circle S 1 . For a 

smooth map fÏ MP denote by f the set of its critical points. Let also S( f )h¾ 

D(M)Ï fÆh f be the stabilizer of f with respect to the action of D(M) on C½ (M, P) 

and S id ( f ) be the identity component of S( f ) with respect to the W½ -topology. 

Theorem 3.7 (cf. [15, Theorem 1.3]). Let f ¾ C½ (M, P). Suppose that 

(i) f takes constant values on connected components of M and has no critical points 

on M;


(ii) for every z¾ f the map f is C½ equivalent near z to a homogeneous polynomial 

g z without multiple factors such that deg g z 2. 

If M is orientable and f consists of non-degenerate local extremes only, i.e. f 

is a Morse map without critical points of index 1, then S id ( f ) is homotopy equivalent 

to the circle. In all other cases S id ( f ) is contractible as well. 

Sketch of proof. It suffices to assume that M is orientable. A non-orientable case 

will follow from an orientable one by arguments of [15, §4.7]. 

Similarly to [15, Lemma 5.1] using (i) and (ii) it is possible to construct a vector 

field F on M with the following properties: 

(A) d f (F)0, in particular F is tangent to M; 

(B) F(z)0 if and only if z¾M is a critical point of F, i.e.F f ; 

(C) for every critical point z¾ f there exists a local presentation fÏÊ2Ê of f 

in which z 0, f is a homogeneous polynomial of degree 2 and without multiple 

factors, and F(x, y)( f¼y , f ¼x ) is a Hamiltonian vector field of f . 

Then it follows from (A), (B), and arguments of [15, Lemma 3.5] that S id ( f )D id (F)½ . 

Notice that F belongs to class F(M). Indeed, conditions (a) and (b) are evident. 

Moreover, for each periodic point of F its first recurrent map is the identity, which 

implies (c). 

Finally by (C) each non-degenerate saddle of f is of type (L1), each non-degenerate 

local extreme of f is of type (L4), while all degenerate critical points of f are of types 

(H). This implies (d). 

Hence S id ( f )D id (F)½ is either contractible or homotopy equivalent to S 1 . If M 

is orientable and f 

consists of non-degenerate local extremes only, then f is Morse 

and belongs to one of the types (A)–(D) of [15, Theorem 1.9]. In this case the corresponding 

shift map is not injective, whence S id ( f ) is homotopy equivalent to S 1 . 

In all other cases ³ is injective and so S id ( f ) is contractible. Indeed, if f has a 

saddle critical point then F has a on-closed orbit, while if f has a degenerate local 

extreme z then j 1 F(z)0. 

4. Openness of ³ 

Let V M be a D-submanifold. Our aim is to find sufficient conditions for the 

shift map ³VÏ func(F, V )Sh(F, V ) to be a local homeomorphism with respect to 

S½ -topologies. Since ³V is S r,r -continuous for all r¾Æ0, this is equivalent to S½,½ - 

openness of³V . Moreover, as³V 

is locally injective, we can also require that its local 

inverses are S½,½ -continuous and defined on S 

½ -open subsets of Sh(F, V ). 

In this section we show that S r,s -openness of³V for some r, s¾Æ0 is equivalent to 

S s,r -continuity of its local inverse defined only on some neighbourhood of the identity 

inclusion i VÏ V M in Sh(F, V ).


DEFINITION 4.1. Let A be a group and S be a semigroup with unit e. Then 

the right action of S on A is a map £Ï A¢ S A such that «£ e « and «£ 

( f g)(«£ f )£g for all «¾ A and f , g¾S. A map Ï S A is called a crossed 

homomorphism if 

( f g)( ( f )£ g) (g). 

Suppose for the moment that F generates a global flow F. Put S Sh(F) and 

Afunc(F)C½ (M,Ê). Then A is an abelian group and by [13, Equation (8)] S is 

a subsemigroup of C½ (M, M) acting from the right on A as follows: 

«Æ sÏ s M M 

«Ê, «¾ A, s¾S. 

Suppose also that the shift map ³ of F is injective, i.e., ³Ï C½ (M,Ê)Sh(F) is a 

bijection. Then it follows from [13, Equation (8)] that the inverse map 

³ 1Ï Sh(F)C½ (M,Ê) 

is a crossed homomorphism, i.e. 

( f Æ g) ( f )Æg (g), 

f , g¾S. 

If ³ is not injective, then the local inverse of ³ near id M is a “local crossed homomorphism”. 

Moreover, if F is not a global flow, then func(F) is an open subset of 

C½ (M,Ê) containing the zero function, i.e. a “local group”. 

Notice that if Sh(F) were a group and³ 1 were a homomorphism, then continuity 

of ³ 1 would be equivalent to its continuity at the unit element id M only, which of 

course is a simpler problem. In our case Sh(F) is just a semigroup, ³ 1 is a local 

crossed homomorphism, and func(F, V ) is in general a local group. Nevertheless we 

will now show that continuity of local inverses of ³ is equivalent to continuity of the 

local inverse of ³ at i VÏ V M. 

Theorem 4.2. Let V M be a D-submanifold and r, s¾Æ0. Then the following 

conditions are equivalent: 

(1) The shift map ³VÏ func(F, V )Sh(F, V ) is S r,s -open; 

(2) For every «¾ func(F, V ) there exists an S s -neighbourhood N f of f ³V («) in 

Sh(F, V ) and an S s,r -continuous section of ³V , i.e. a map 

Ï N f C½ (V ,Ê), 

such that ( f )« and ³V Æ 

for all g¾ N f and x¾ V . 

id(N f ). In other words 

f (x)F(x, ( f )(x)),


(3) Property (2) holds for the zero function « 0Ï V Ê and the identity inclusion 

f ³V (0)i VÏ V M. 

Proof. (1)µ(2). Suppose that³VÏ func(F, V )Sh(F, V ) is S r,s -open. Since 

³V is locally injective with respect to the S 0 -topology, condition (1) means that for every 

«¾ func(F, V ) there exists an S r -open neighbourhood M« in func(F, V ) such that 

a) the restriction ³VM«Ï M«³V (M«) is a bijection, 

b) ³V (M«) is S s -open in Sh(F, V ), so ³V (M«)Sh(F, V )N f , for some S s -open 

N f 

1 

V 

neighbourhood of f ³V («) in C½ (V , M), 

Ï³V (M«)M« is S s,r -continuous. 

c) the inverse map V ³ 

def 

Then N f ³V (M«)Sh(F, V )N f and V satisfy condition (2). 

The implication (2)µ(3) is evident. 

(3) µ (1). It suffices to show that for every « ¾ func(F, V ) there exists an 

S r -open neighbourhood M« in func(F, V ) satisfying conditions a)–c) above. Condition 

(3) means that such a neighbourhood M 0 exists for the zero function 0. 

For every «¾ C½ (V ,Ê) define the following subset of C½ (V , M) 

U«f ¾ C½ (V , M)Ï ( f (x),«(x))¾dom(F),x¾ V 

and consider a map q«Ï U« C½ (V , M) defined by 

q«( f )(x)F( f (x),«(x)), f ¾ U«, x¾ V . 

It is easy to see that U« is S 0 -open in C½ (V , M). Also notice that, q « is S r,r -continuous 

for all r ¾Æ0 and that i V ¾ U« if and only if«¾ func(F, V ). 

Lemma 4.3. The image of q« coincides with U « and q « is its inverse, i.e., 

1 

q « q « Ï U « U«. Thus q« is an S r,r -homeomorphism (r ¾Æ0) between the 

S 0 -open sets U« and U «. 

Proof. 

Let f ¾ U«, i.e. ( f (x),«(x))¾dom(F) for all x¾ V . Then 

(q«( f )(x), «(x))(F( f (x),«(x)), «(x))¾dom(F), 

q «Æ q«( f )(x)F(F( f (x),«(x)), «(x))F( f (x),«(x) «(x)) f (x). 

Hence q«(U«)U « and q «Æq« id U« . Interchanging« and « will give the result. 

Now we turn to the proof of Theorem 4.2. For every «¾ func(F, V ) set 

M« 

def 

(M 0³ 

1 

Then M« is S r -open in func(F, V ). 

V (U «)«)func(F, V ).

0¾³ 

1 

V (i V )³ 

1 

V (U) and thus «¾ M «. 

Moreover «¾ M«. Indeed, since «¾ func(F, V ), we have that i V ¾ 

We will show that M« satisfies the conditions a)–c) above. 

a) The restriction of ³V to M« is 1-1, since so is ³VM 0 

. 

Conditions b) and c) are implied by the following lemma: 

U«, whence 


Lemma 4.4. 

in Sh(F, V ). 

Let M¼« M« be any S r -open subset. Then ³V (M¼« 

) is Ss -open 

Proof. Denote M¼0M¼« «. Then M¼0M 0³ 

1 

V (U «) and by the condition b) 

for M 0 there exists an S s -open subset N¼ of C½ (V , M) such that 

³V (M¼0 )Sh(F, V )N ¼ . 

Define the following S r,r -homeomorphism 

a«Ï C½ (V ,Ê)C½ (V ,Ê), 

Then we obtain the following commutative diagram: 

(4.1) 

M¼0 

³V 

Ã 

M¼« 

³V 

a«Ã 

a«(¬)«¬. 

³V (M¼0 

¼U« 

)Sh(F, 

q« 

V )N 

Ã Sh(F, V )q«(N¼U«) 

Ã 

simply meaning that F(x,¬(x)«(x))F(F(x,¬(x)),«(x)) for all¬¾M 0 and x¾ V . 

We claim that 

³V (M¼« )Sh(F, V )q «(N¼U«). 

Indeed, by (4.1), ³V (M¼« )Sh(F, V )q «(N¼U«). 

Conversely, let g ¾ Sh(F, V )q«(N¼ U«), i.e., there exist ¬ ¾ func(F, V ) and 

f ¾ N¼U« such that 

(4.2) g(x)F(x,¬(x))F( f (x),«(x)), x¾ V . 

Then f (x)F(x,¬(x) «(x)), i.e. f ¾ Sh(F, V ) and therefore 

f ³V (¬ «)¾Sh(F, V )N¼U«³V (M¼0 ). 

Thus there exists ¾ ¬ M¼0 

( , possibly distinct from «, such that f ³V ). Denote 

¬¼ «. Then ¬¼¾ M¼« and 

g(x)F( f (x),«(x))F(F(x, (x)),«(x))F(x,¬¼ (x)).


In other words, g³V (¬¼ )¾³V (M¼« ). Lemma 4.4 and Theorem 4.2 are completed. 

5. Examples when shift map is not open 

In this section we discuss four examples of well-known flows whose shift maps turn 

out not to be homeomorphisms onto their images. They have similar nature, but are 

given on different types of manifolds. These examples also provide counterexamples to 

[13, Theorem 1]. All manifolds in this section are compact, therefore we will not distinguish 

weak and strong topologies. In all the examples below our vector fields will 

satisfy the assumptions of the following simple lemma: 

Lemma 5.1. Let F be a vector field on a compact manifold M tangent to M. 

Suppose that the shift map³ of F is injective and for every r¾Æ0, a W r -neighbourhood 

N of id M , and arbitrary large T 0 there exists t ¾Ê, regarded as a constant function 

tÏ M Ê, such that tT and F t ³(t)¾N . Then ³ is not W r,s -open for 

any r, s¾Æ0. In particular, ³ is not a homeomorphism onto its image with respect to 

W½ -topologies of C ½ (M,Ê) and Sh(F). 


Consider the following W 0 -neighbourhood of the zero function: 

M 0«¾ C½ (M,Ê)Ï«(x)1. 

Suppose that there exists a W r -neighbourhood N of id M 

assumption there exists a constant function t 1 such that³(t)¾N . Since³ is injective, 

and tM 0 , we obtain that³(t)³(M 0 ), whence Nê³(M 0 ). This contradiction 

such that N ³(M 0 ). By 

implies that ³ is not W r,s -open for any r, s¾Æ0. 

Irrational flow. For simplicity we will consider the irrational flow on T 2 . Let 

¾Ê be an irrational number and F be the irrational flow on the 2-torus T 2Ê22 

given by F(x, y, t)(x t, y t). 

Lemma 5.2. 

for any r, s¾Æ0. 

The shift map ³Ï C½ (T 2 ,Ê)C½ 

(T 2 , T 2 ) of F is not W r,s -open 

Proof. Notice that every orbit of F is non-closed and everywhere dense, so ³ is 

injective. First we give convenient formulas for the metrics generating W r -topologies 

on C½ (T 2 , T 2 ). Let fÏ T 2T 2 be a C½ map. Then f lifts to some 2 -equivariant 

map f É (É f1 , f2 É 2 )ÏÊ2Ê2 . Let I [0, 1]¢[0, 1]Ê2 be the fundamental domain 

. Define the r-th norm of f by 

for the covering map pÏÊ2T 2 

fr 

 

sup min(É f j, 1 É f 

(x, y)¾I 

j) 

2 

j1,2 

1i 1i 2r 

sup 

(x, y)¾I 2¬ 

i 1i 2 f É j 

x 1 i y i 2 

¬ ,


be a base W r -neighbourhood 

of id T 2 ³(0)¾C½ (T 2 , T 2 ) for some0. We will show that there exists arbitrary 

large (by absolute value) n ¾ such that ³(n)F n ¾ N r . Then our lemma will 

where t is the fractional part of the absolute value of t¾Ê. 

Let r 0 and N f ¾C½ (T 2 , T )Ï 2 f id T 2r 

follow from Lemma 5.1. 

Notice that F n(x, y)(xn, yn)(xn, y) for all n¾, i.e., F n is just 

a “rotation along the first coordinate”. Since this map is defined by adding constants to 

coordinates, it follows from formula for fr that for each r¾Æ0 the distance between 

id T 2 and F n with respect to the W r -topology is equal to 

F n id T 2r 

 

min(n, 1 n) 

and therefore does not depend on r. 

Sinceis irrational, the set Tmin(n, 1 n)n¾ is everywhere dense 

in S 1 

Ê. Hence there are arbitrary large (by absolute value) n ¾ such that 

F n id T 2r 

, i.e. F n¾ r N . 

be the unit disk 

Irrational flows on a solid torus. Let D z¾Ïz1 2 

in the complex plane, T S 1¢D2 be the solid torus, and be an irrational number. 

Define the following flow on T by F(, z, t)( 2t t mod 1, e z). Then by 

the arguments similar to Lemma 5.2 is can be shown that F satisfies assumptions of 

Lemma 5.1. 

The main feature of this example is that F has periodic orbit 1¢ S 0 and all 

other ones are recurrent. Let B 1¢ D 2 be and RÏ BB be the first recurrence map 

of 2 defined by R(z)e z, i.e. it is the rotation by 2. Then R is not periodic, 

eigen values of its tangent map T 0 R at 0¾ B have modulus 1, and the iterations of R 

can be arbitrary close to id B . Thus F satisfies all assumptions but (b) of Definition 3.4 

of class F(T 2 ). 

Periodic linear flows. 

Ê2nn by 

Given 1,,n¾Ê define the following linear flow on 

(5.1) F(z 1 ,, z n , t)(e i1t z 1 ,, e int z n ). 

Evidently, the closed 2n-disk D r of radius r and centered at the origin is invariant with 

respect to F. 

Lemma 5.3. 

The following conditions are equivalent: 

(1) F idn for some 0, i.e. j ¾ for all j 1,, n; 

(2) every z¾n is either fixed or periodic with respect to F; 

(3) at least one point z (z 1 ,, 

z n ) with all non-zero coordinates is periodic. 

A flow satisfying one of these conditions will be called periodic.



i The implications (1)µ(2)µ(3) are evident. (3)µ(1) Let z¾n be a 

periodic point with all non-zero coordinates and be the period of z with respect to 

F. Then e j z j z j 0 for all j 1,, n, whence j¡2 ¾. 

Lemma 5.4. 

W r,s -open for any r, s¾Æ0. 

If F is not periodic then its shift map ³ is injective and is not 

Proof. By Lemma 5.3 every point z¾n with all non-zero coordinates is nonperiodic 

and it is easy to see that its orbit is dense on some open subset of the sphere 

S 2n 1 

x Dx of radius x. In particular, this orbit is non-closed, whence the shift 

map of F is injective. Then similarly to Lemma 5.2, we can find arbitrary large t¾Ê 

such that F t is arbitrary close to idÊ2n in any of W r -topologies. 

Flow on S 2n , n2. We will now extend the last result for the construction of a 

flow on the sphere S 2n with two fixed points at which this flow is linear. 

Let Ê2n 

1 

and Ê2n 

2 

be two copies of Ê2n n . Define a diffeomorphism ÏÊ2n 

1 

Ò 

0Ê2n 

2 

Ò0 by(z)zz2 , wherez is the usual Euclidean norm inÊ2n . Then 

maps every sphere of radius r centered at 0 to the sphere of radius 1r. Gluing Ê2n 

1 

and Ê2n 

2 

via we obtain a 2n-sphere. 

Notice that Equation (5.1) defines the flows on Ê2n 

1 

and Ê2n 

2 

so that the following 

diagram is commutative: 

Ê2n 

1 

Ò0 

F tÃ 

Ê2n 

1 

Ò0 

Ã 

 

ÃÊ2n 

Ê2n 

2 

Ò0 

F t 

Ã 

2 

Ò0. 

Hence these flows determine a unique flow F¼ 

on S 2n with two fixed points. Moreover, 

F¼ 

is linear on the charts Ê2n 

1 

and Ê2n 

2 

at these points. 

Now suppose that F is not periodic. Then we can find arbitrary large t ¾Ê such 

that F t is arbitrary close to idÊ2n in any of W r -topologies. This implies that F¼ 

satisfies 

assumptions of Lemma 5.1. 

6. Regular and trivial extensions 

The results of this section will allow to estimate continuity of shift maps for vector 

fields of types (L) and (H). 

Let M, N be two manifolds, G be a vector field on M, F be some regular extension, 

and Æ G be the trivial extension of G on M¢N. Thus 

F(x, y)(G(x), H(x, y)), Æ G(x, y)(G(x), 0),


for some smooth map HÏ M¢N T N. By F, Æ G, and G we will denote the corresponding 

local flows, and by ³, Æ, and the corresponding shift maps of F, Æ G, and 

G. It is easy to see that 

(6.1) dom(F)dom(Æ G)dom(G)¢ N. 

Moreover, 

(6.2) F(x, y, t)(G(x, t), H(x, y, t)), Æ G(x, y, t)(G(x, t), y), 

for some smooth map HÏ dom(Æ G) N. 

Let V M¢N be a connected compact D-submanifold. Then it follows from (6.1) 

that func(F, V ) is W 0 -open in func(Æ G, V ). Define the map 

PÏ Sh(F, V )Sh(Æ G, V ) 

by the rule: if f ¾ Sh(F, V ), and f (x, y) (A(x, y), g(x, y)), then P( f )(x, y) 

(A(x, y), y). It follows from (6.2) that P is well defined and is W r,r -continuous for 

every r ¾Æ0. Moreover, we have the following commutative diagram: 

(6.3) 

Theorem 6.1. 

³V 

func(F, V ) Ã Sh(F, V ) 

ÆV 

P 

ÆV 

func(Æ G, V ) Ã Sh(Æ G, V ). 

¸ Ã 

Ã 

Ã 

Suppose thatG is nowhere dense in M and that ÆV is W r,s -open 

for some r, s¾Æ0. 

(1) If ÆV is injective, then ³V is W r,s -open as well. 

(2) If ³V is W r,s -open, then P is locally injective with respect to the W s -topology 

of Sh(F, V ). 

Suppose in addition that ÆV is also W s,t -open for some t¾Æ0. 

(3) If P is locally injective with respect to the W s -topology of Sh(F, V ), then ³V is 

W r,t -open. 

(4) If both ÆV and³V are not injective, then P is locally injective with respect to the 

W 0 -topology of Sh(F, V ), whence, by (3), ³V is W r,t -open. 

Proof. (1) If ÆVÏ func(Æ G, V )Sh(Æ G, V ) is a bijection, it follows from (6.3) 

that so is ³V . Let Mfunc(F, V ) be a W r -open subset. Then 

³V (M) P 

1Æ ÆV (M) 

is W s -open in Sh(F, V ) due to W r,s -openness of ÆV and W s,s -continuity of P.


(2) Let «¾ func(F, V ) and f ³V («). We will find a W s -neighbourhood of f 

such that PN is 1-1. 

Since both³V and ÆV are W r,s -open and locally injective with respect to W 0 -topologies 

of func(F, V ) and func(Æ G, V ) respectively, there exists a W r -neighbourhood M of« in 

func(F, V ) such thatN ³V (M) is a W s -neighbourhood of f in Sh(F, V ) and the restrictions 

of³V and ÆV to M are 1-1. Then it follows from (6.3) that PN is 1-1 as well. 

(3) Let Mfunc(F, V ) be a W r -open subset and«¾M. We will show that there 

exists a W s -neighbourhood M¼ M of « such that ³V (M¼ ) is W t -open in Sh(F, V ). 

Since«¾ M is arbitrary, we will obtain that³V (M) is W t -open in Sh(F, V ). 

It follows from W r,s -openness of ÆVM and W s,s -continuity of P that the set P 1Æ 

ÆV (M) is a W s -open neighbourhood of³V («) in Sh(F, V ). Moreover, since P is locally 

injective with respect to the W s -topology of Sh(F, V ), there exists a W s -neighbourhood 

N P 1Æ ÆV (M) of³V («) such that the restriction PNÏ N Sh(Æ G, V ) is injective. 

1 

Denote M¼ M³ 

V 

(N ). We claim that 

³V (M¼ ) P 

1Æ ÆV (M¼ )N , 

whence ³V (M¼ ) will be W t -open in Sh(F, V ). Indeed, 

P 

1Æ ÆV (M¼ )N (6.3) 

injectivity of PN 

P 

1Æ PÆ³V (M¼ )N ³V (M¼ ). 

(4) Suppose that both ÆV and ³V are not injective. We will show that for some 

m 0 there exists a free P-equivariant action of the group m on Sh(F, V ) such that 

P( f ) P(g) if and only if f , g belongs to the same m-orbit. This will give us 

a decomposition 

PÏ Sh(F, V ) q Sh(F, V )m Sh(Æ G, V ) 

in which q is a covering map, and is a bijection with the image P(Sh(F, V )) 

Sh(Æ G, V ). Then q will be locally injective with respect to the W 0 -topology since m 

is a finite group. Hence so will be P. 

By Lemma 2.3 (5b) func(Æ G, V )func(F, V )C½ (V ,Ê), ker(ÆV )nÆn¾, and 

ker(³V )nn¾ for some smooth functions Æ,Ï M¢N (0,½) such that 

ÆG(x, y, tÆ(x, y)) Æ G(x, y, t), 

F(x, y, t(x, y))F(x, y, t). 

In other words 

(6.4) 

(G(x, tÆ), y)(G(x, t), y), 

(G(x, t), H(x, y, t))(G(x, t), H(x, y, t)). 

It follows that Æ G(x, y, t)(G(x, t), y)(G(x, t), y) Æ G(x, y, t), whence 

¾ker(ÆV ), i.e.mÆ for some m¾.


In particular, by (2) of Lemma 2.3, Æ is constant along orbits of F. Since for every 

«¾ func(F, V ) the map ³V («) preserves every orbit of F, we have that ÆÆ³V («)Æ, 

whence 

[13, Equation (8)] 

³V (Æ)Æ³V («) ³V («ÆÆ³V («))³V («Æ). 

Then we can define an action £ of the group m on Sh(F, V ) by 

k£ f ³V (kÆ)Æ f , k¾m, f ¾ Sh(F, V ). 

Evidently this action is free and by (6.4) is equivariant with respect to P. 

Moreover, let f ³V («), g ³V 

(¬) for some «,¬ ¾ func(F, V ) and suppose 

that P( f ) P(g), i.e. ÆV («) ÆV (¬). Then it follows from (3) of Lemma 2.3 that 

« ¬ kÆ for some k ¾, whence f ³V (kÆ)Æg. Moreover, since mÆ, we 

may take k modulo m. In other words, P( f ) P(g) if and only if f k£ g for some 

k¾m. 

7. Vector fields of types (L) and (H) 

The following lemma shows that property of openness of shift maps near singular 

points is invariant under reparametrizations. 

Lemma 7.1 (cf. [20].). Let F be a C½ vector field on M, Ï M (0,½) a 

C½ strictly positive function, and G F. Let also V M be a D-submanifold and 

³V andV be shift maps for F and G respectively. Then Sh(F, V )Sh(G, V ). Moreover, 

³V is S r,s -open iff so is V . 


Define the functions «Ï dom(F)Ê and ¬Ï dom(G)Ê by 

¬(x, s) s 

0 0 

(G(x,)) d, «(x, s) s 

d 

(F(x,)) . 

Then it is well-known that for each f ¾ func(F, V ) and g¾ func(G, V ), see e.g. [20], 

G(x, g(x))F(x,¬(x, g(x))), 

F(x, f (x))G(x,«(x, f (x))), 

for all x¾ V . Define the following map 

Ï func(G, V )func(F, V ), (g)(x)¬(x, g(x)). 

Then is a homeomorphism with respect to topologies W r for all r, and its inverse is 

given by 1 ( f )(x)«(x, f (x)). Moreover V ³V Æ . Hence ³V is S r,s -open iff 

so is V .


Thus for the study of vector fields of types (L) and (H) is suffices to consider 

linear and reduced Hamiltonian vector fields only. 

Lemma 7.2 ([19, 14]). Let gÏÊ2Ê be a homogeneous polynomial, G be its 

reduced Hamiltonian vector field, F be a trivial n-extension of G on Ên¢Ê2 , and V 

of F is W½,½ -open. If 

be a D-neighbourhood of 0¾Ên2 . Then the shift map ³V 

deg G 2, i.e. G is of type (H), then for every regular n-extension of G its shift map 

is W½,½ -open as well. 

Proof. It follows from [19, 14] that the shift map ÆV of G Æ has a W 

½,½ -continuous 

local section on some neighbourhood of the identity inclusion i VÏ V Ên2 , see also 

[19, Theorem 11.1]. By Theorem 4.2 this implies W½,½ -openness ofV . 

Moreover, if deg G 2, i.e. the linear part of F at 0 vanishes, then by Lemma 2.3 

³V is injective. Hence by (1) of Theorem 6.1 the shift map of G is W½,½ -open as well. 

Lemma 7.3. 

Let B be a non-zero (k¢k)-matrix, G(y) By be the corresponding 

linear vector field on Êk , G(y, t)e Bt y be its flow, and F be a vector field on Ênk 

being a regular (possibly trivial) n-extension of G with respect to the origins of Ênk 

and Êk as indicated in the third column of the table below. Let also V Ênk be a 

D-neighbourhood of the origin 0¾Ênk . Then the shift map³V of F is W r,s -open for 

the values r, s described in the following table. 

type B F openness of ³V 

1 (L1) Ba, a 0, regular W r,r1 , r 0 

2 (L1) B a b , a, b0 regular W 

b a 

r,r2 , r 0 

3 (L2) B 0 1 

0 0, regular W r,r1 , r 0 

4 (L3) B 0 b 1 0 

b 0 0 1 

5 (L4) B 0 b 

6 B 

6a 

(L4) 

 

, b0 regular W½,½ 

0 0 0 b 

0 0 b 0 

, b0 trivial W½,½ 

b 0 

0 b 1 

b 1 0 

 

, l 2 

. .. 

0 b l 

b l 0 

(a) b j¾ for some 0 and all 

j 1,, l (periodic case) 

trivial 

W½,½ 

6b (L5) (b) otherwise (non-periodic case) — —


Proof. In Cases 1–3 we will first suppose that F is a trivial n-extension of G. 

Then F is linear and is generated by the following matrix 0 n 0, where 0 

0 B 

n is the 

zero (n¢ n)-matrix. Let denote the coordinates in Ên . 

CASE 1. In this case F(, x) axx, (a 0), F generates the following 

global flow F(, x, t)(, xe at ) on Ên1 , ³V is injective (because F has non-closed 

orbits), its image Sh(F, V ) consists of maps f ( f 1 , f 2 )¾C½ (V ,Ên¢Ê) satisfying 

the following conditions: 

f 1, f 2 

f 2 (, 0) 

(, 0)0, 0, x f 2 (, x)0 (x 0), 

x 

Ï Sh(F, V )C½ (V ,Ê) is given by 

and the inverse mapping ³ 

1 

V 

(7.1) ³ 

1 

V ( f )(, x) 1 a ln f 2(, x) 

x 

1 a ln 1 

0 

f 2 (, tx) 

x 

1 

see [13, Equation (23) and (27)]. Hence³ 

V 

is Wr1,r -continuous for all r 0, whence 

³V is W r,r1 -open. 

CASE 2. We will regard Ên2 as Ên¢. Since a 0, we have that F has nonclosed 

orbits, whence again its shift map ³V is injective but now the image Sh(F, V ) 

Æ f2 

can not be described so simply as in the previous cases. Let «¾ C½ (V ,Ê) and f 

( f 1 , f 2 )³V («)¾C½ (V ,Ên¢). Then f 1. Notice that we can define complex 

and its partial derivatives f2z Æ and f2Æz Æ in z and Æz in a usual way. 

conjugate 

Then it follows from [13, Equation (29) and Lemma 34] that 

«(, z) Im( 1 

2a ln f 2 d f2 Æ (Æ)) 

1 

Im(zÆ) 

dt, 

Im(¡ 2¡ (( 

2a 

f2z)( Æ Æ ln f 

2 f2Æz)Æ)) 

y(a b 2 ) 

and that the numerator of the last fraction is equal to zero, when y 0. It follows 

from this formula and the Hadamard lemma that the expression of « via f contains 

1 

partial derivatives of f up to order 2. Hence ³ 

V 

is W r2,r -continuous for all r 0. 

CASE 3. Now F(, x, y) yx for (, x, y)¾Ên2 and it generates the following 

flow F(, x, y, t)(, x yt, y). Then³V is again injective (since F has non-closed 

orbits), its image Sh(F, V ) consists of mappings f ( f 1 , f 2 , f 3 )¾C½ (V ,Ên¢Ê¢Ê) 

such that 

and the inverse map ³ 

1 

V 

Ï Sh(F, V )C½ (V ,Ê) is given by 

f 1, f 2 (, x, 0)0, f 3y, 

(7.2) ³ 

1 

V ( f ) f 2(, x, y) x 

y 

1 

0 

f 2 (, x, ty) 

dt, 

y 

see [13, Equation (26)]. Hence ³ 

1 

V 

is W r1,r -continuous for all r 0.


Since in Cases 1–3 ³V is injective, it follows from (1) of Theorem 6.1 that the 

1 

same estimations of continuity of ³ 

V 

hold for regular extensions of G. 

Notice that in the remaining Cases 4–6 F is a regular n-extension of the linear 

vector field G 5 byx bx y defined by the matrix B 5 0 b 

. In fact, it 

b 0 

is easy to see that this vector field is the Hamiltonian vector field of the homogeneous 

polynomial g(x, y)(b2)(x 2y2 ). Also notice that the shift map ³V of any trivial 

extension F of G 5 is not injective and its kernel ker(³V ) consists of integer multiples 

of constant function 2b. 

CASE 5. It follows from Lemma 7.2 that local inverses of³V are W½,½ -continuous. 

An independent proof of W½,½ -continuity of³V for this case is also given in [21] under 

more general settings. Notice that in this case we claim nothing about openness of shift 

maps of regular extensions F of G 5 , since due to Theorem 6.1 it is necessary to have 

additional information about F. 

CASE 4. We will regardÊ4 as2 . Suppose at first that F is a trivial n-extension 

of G. Since G is a regular 2-extension of G 5 , we see that F is a regular (n 2)- 

extension of G 5 . Let also Æ G be a trivial (n 2)-extension of G 5 . Thus F and Æ G are 

defined on Ên¢¢ and generate the following global flows: 

F(, z 1 , z 2 , t)(, e ibt (z 1 tz 2 ), e ibt z 2 ), Æ G(, z 1 , z 2 , t)(, z 1 , e ibt z 2 ). 

Denote 2b and let V Ên¢2 be a D-neighbourhood of 0¾Ên¢2 . For 

every «¾ C½ (V ,Ê) put 

f(«)(, z 1 , z 2 )e ib«(,z 1 ,z 2 ) z 1 , g(«)(, z 1 , z 2 )e ib«(,z 1 ,z 2 ) z 2 , 

Then the shift maps ³V and ÆV of F and G respectively are given by 

(7.3) ³V («)(, f(«)«¡ g(«), g(«)), ÆV («)(, z 1 , g(«)). 

Similarly to (6.3) define PÏ Sh(F, V )Sh(Æ G, V ) by the following rule: if h 

(, f , g)¾Sh(F, V )C½ (V ,Ên 

¢¢), then P(h)(, z 1 , z 2 )(, z 1 , g(z 1 , z 2 )). 

Evidently, ÆV PÆ³V . 

By Case 5 the shift map ÆVÏ C½ (V ,Ê)Sh(Æ G, V ) is W 

½,½ -open. We claim 

that so is ³VÏ C½ (V ,Ê)Sh(F, V ). Since G has non-closed orbits, ÆV is always 

injective, whence it will follow from (1) of Theorem 6.1 that the shift map of any 

regular extension of G is W½,½ -open as well. 

By (3) of Theorem 6.1 it suffices to show that P is locally injective with respect 

to the W 1 1 

-topology. Evidently, 

V 

ÆV («)«nn¾, whence we obtain from 

(7.3) that 

P 

1Æ ÆV («)³V («)(0,n¡ g(«), 0)n¾.


Now, define the following CW 1 -neighbourhood of ³V («) in Sh(F, V ): 

 

. 

N f ¾ Sh(F, V ) 

We claim that the restriction PN is 1-1. 

and ³V (¬n) 

Then 

f ³V («)1 

V 

 

4 

Since³V is injective map, it suffices to establish that n 0, whenever both³V (¬) 

belong to N for some ¬¾ ib¬ C½ (V ,Ê). Notice that 

g(¬)1 

V g(¬)¼z 2(z 1 ,z 2 )0e (ibz2¬¼z 2 1)(z 1 ,z 2 )0 1. 

³V (¬) ³V (¬n)1 

V ³V (¬) ³V («)1 

V ³V (¬n) ³V 

 

2 . 

On the other hand, 

³V (¬) ³V (¬n)1 

V (0,ng(¬), 0)1 

V n. 

(«)1 

V 

Hence n 0. 

CASE 6a. In this case F is a regular extension of G 5 . Since the flow F is periodic, 

we have that the shift maps of F and G 5 are not injective and by Case 5 the 

shift map of G 5 is W½,½ -open. Then by (4) of Theorem 6.1 so is the shift map of F. 

CASE 6b. In this case³V is injective, but as it is shown in Lemma 5.4 its inverse 

map is not even W½,½ -continuous. 

Lemma 7.3 is completed. 

REMARK 7.4. Incorrect estimations of continuity of local inverses of³V given in 

[13, pp. 199–200] were bases on at the following “division lemma”, which was wrongly 

formulated 1 in [13]. 

Lemma 7.5 (cf. [13, Lemma 32]). Let be either Ê or , V be a D- 

submanifold, and ZÏ C½ (V ,)C½ (V ,) be a map defined by the formula: Z(«)(x) 

x¡«(x). IfÊ then the inverse map Z 1Ï im Z C½ (V ,) is W r1,r -continuous 

for all r 0. If, then Z 1 is only W½,½ -continuous. 

The caseÊ easily follows from the Hadamard lemma, see also [25]. The case 

is more complicated and can be established by methods of [19, 14] but during 

the proof of Lemma 7.3 we avoided referring to it. 

1 In [13, Lemma 32] it was claimed that Z 1 is W r,r -continuous for all r ¾Æ0. But the latter 

inequality in the proof of [13, Lemma 32] actually shows W r,r -continuity of Z but not of its inverse.


8. Fragmentation 

The aim of this section is to repair [13, Theorem 17] by giving a sufficient condition 

for the shift map³Ï func(F)Sh(F) to be either a homeomorphism or an infinite 

cyclic covering map with respect to S½ -topologies, see Theorem 8.2. 

REMARK 8.1. The error occurred in the third paragraph of [13, Theorem 17], 

where it was claimed that “. . . the image N i ³U i (M¼i ) is a Cr W 

-neighbourhood of 

fU i 

in C½ (Ui , M) for all r ¾Æ0.” First of all this phrase contains a misprint: instead 

of C½ (Ui , M) the author supposed to be written im³U i 

. But nevertheless the 

statement that N i is CW r -open in im³U i 

does not follow from the assumptions of [13, 

Theorem 17] and must be included into the formulation of “section” property (S) 0 of 

[13, Definition 15]. This is the point which was missed. 

Theorem 8.2 (cf. [13, Theorem 17]). 

Let F be a vector field on M such that 

F is nowhere dense. Suppose that there exists a¾Æ0, a function dÏÆ0Æ0, a locally 

finite coverV ii¾¿ of M by D-submanifolds, and a finite (possibly empty) subset 

¿¼¿ such that Int V ii¾¿ is also a cover of M, and the local shift map 

³V iÏ 

func(F, V i )Sh(F, V i ) 

is S½,½ -open if i¾¿¼ and S r,d(r) -open for all r a if i¾¿Ò¿¼ . Then the shift map 

³Ï func(F)Sh(F) 

is S½,½ -open. Moreover, if ¿¼û, then ³ is S r,d(r) -open for every r a. 

Hence if³Ï func(F)Sh(F) is injective, then it is a homeomorphism with respect 

to S½ -topologies. Otherwise, ³ is a -covering map. 

Proof. The proof follows the line of [13, Theorem 17]. Since F is nowhere 

dense, there exists a continuous function ÆÏ M (0,½) such that for every periodic 

point x of period x we have that Æ(x)3x, see [13, Proposition 14]. 

Let«¾ func(F), r a, d r be a metric on the manifold J r (M,Ê) of r-jets,Ï M 

(0, 1) be a strictly positive continuous function such that Æ, and 

M¬¾ def 

func(F)d r ( j r«(x), j x¾M 

r¬(x))(x), 

be a base S r -neighbourhood of «, where j r¬(x) denotes the r-jet of ¬ at x. Then the 

restriction of ³ to M is injective, [13, Proposition 14]. We will show that ³(M) is 

S½ -open in Sh(F) and if ¿¼û, then ³(M) is even S d(r) -open in Sh(F). This will 

complete Theorem 8.2. 

For every i¾¿ let 

def 

(8.1) M i ¬¾ func(F, V i )Ï d r ( j r«(x), j x¾ r¬(x))(x), V i

p iÃ 


in func(F, V i ). It follows from assumption about open- 

be an S r -neighbourhood of«V i 

ness of ³V i 

that 

³V i 

(M i )Sh(F, V i )N i , 

where N i is S½ -open in C ½ (Vi , M) for i¾¿¼ and S d(r) -open for i¾¿Ò¿¼ . Moreover, 

the restriction of ³V i 

to M i is one-to-one. 

Let p iÏ C½ (M,Ê)C½ (Vi ,Ê) and q iÏ C½ (M, M)C½ (Vi , M) be the “restriction 

to V i ” maps. Then we have the following commutative diagram 

³ 

func(F) Ã Sh(F) 

(8.2) 

q i 

func(F, V i ) 

³V i 

Ã 

Sh(F, V i ). 

Ã 

By definition ³ and ³V i 

as surjective. It also follows from definitions of func(F) and 

func(F, V i ) and assumption that V i is a D-submanifold, that every ¾ C½ (V ,Ê) extends 

to some Æ ¾ C½ (M,Ê). Moreover, if ¾ func(F, V i ), we can assume that Æ ¾ 

func(F), whence p i (func(F))func(F, V i ). Therefore q i is surjective as well. It follows 

from definition that 

M i¾¿ 

p 1 

i (M i ). 

Put 

(8.3) N def 

i¾¿ 

q 1 

i (N i ). 

SinceV ii¾¿ is a locally finite cover and¿¼ is finite 2 , it follows from [13, Lemma 18] 

that N is S½ -open in C ½ (M, M) and even S d(r) -open if¿¼û. We will now show that 

(8.4) ³(M)Sh(F)N . 

This will complete our theorem. 

It follows from (8.2) and (8.3) that³(M)Sh(F)N . Conversely, let g¾ Sh(F) 

N . Then gV i 

q i (g)¾Sh(F, V i )N i³V i 

(M i ) for all i¾¿ and 

¬¼¾C½ g³(¬¼ ) for some 

(M,Ê). We have to find (possibly another) function¬¾M such that g³(¬). 

Since the restriction of ³V i 

to M i is injective, gV i 

³V i 

(¬i) for a unique ¬i ¾ M i . 

It remains to show that ¬i ¬ j on V i V j for all i, j¾¿. Since Int V ii¾¿ is a 

cover of M as well, the family of functions ¬ii¾¿ will define a unique smooth function 

¬¾Ëi¾¿ p 1 

i 

(M i )M such that ¬V i 

¬i and ³(¬)g. This will prove (8.4) 

and complete our theorem. 

If¿¼ 2 were infinite, then N would be open in the so-called very-strong topology, see [9]


Let x¾ Int V i Int V j for some i, j¾¿. Then g(x)F(x,¬i(x))F(x,¬ j(x)). 

If x is a non-periodic regular point, then ¬i(x)¬ j(x). 

If x is periodic of periodx, then¬i(x) ¬ j(x)bx for some b¾. But¬i(x) 

¬ j(x)«(x) ¬i(x)«(x) ¬ j(x)2Æ(x)x, whence b0. 

Thus ¬i ¬ j on (Int V i Int V j )ÒF. Since F is nowhere dense, we see that 

¬i ¬i 

on all of V i V j as well. 

9. Replacing M with an open subset 

Theorem 8.2 reduces the verification of openness of the shift map to openness of a 

family of shift maps³V i, whereV i is any locally finite cover of M by D-submanifolds. 

Our next aim is to “localize” the verification an openness of³V by replacing all the manifold 

M with an open neighbourhood W of V . 

Let F be a vector field on M, W M be a connected, open subset, and F WÏ W¢ 

Êdom(F W ) W be the local flow generated by the restriction FW of F to W . Then 

dom(F W )dom(F)(W¢Ê) 

and FF W on dom(F W ). Let V W be a D-submanifold. Then func(F W , V ) is an 

S 0 -open subset of func(F, V ), and the corresponding shift map ³W ,V of F W coincides 

with ³V on func(F W , V ), i.e. 

(9.1) ³W ,V ³Vfunc(F W ,V )Ï func(F W , V )Sh(F W , V )Sh(F, V ). 

The following statement characterizes openness of ³V via openness of ³W ,V . 

Theorem 9.1. 

conditions (A)–(C) are equivalent: 

(A) The shift map ³V is S r,s -open; 

Let V W be a D-submanifold, and r, s¾Æ0. Then the following 

(B) The shift map ³W ,V is S r,s -open and its image Sh(F W , V ) is S s -open in Sh(F, V ), 

i.e. there exists an S s -open subset N C½ (V , W ) such that 

(C) The shift map ³W ,V 

Sh(F, V )N Sh(F W , V ). 

is S r,s -open and there exists an S s -neighbourhood N of the 

identity inclusion i VÏ V M in C½ (V , M) such that 

Sh(F, V )N Sh(F W , V ). 

Proof. (A)µ(B). Since func(F W , V ) is an S 0 -open subset of func(F, V ) and³V 

is an S r,s -open map, it follows that the restriction 

³Vfunc(F W ,V )³W ,V


is an S r,s -open map and its image Sh(F W , V ) is S s -open in Sh(F, V ). 

The implication (B)µ(C) is evident. 

(C)µ(A). Suppose that ³W ,V is S r,s -open. Then by (2) of Theorem 4.2 there 

exists an S s -neighbourhood U of the identity inclusion i VÏ V W in C½ (V , W ) and 

an S s,r -continuous section of ³W ,V defined on U¼ U Sh(F W , V ): 

Ï Sh(F, V )Sh(F W , V )U¼ func(F W , V )func(F, V ), 

i.e., ³W ,V Æ id U¼. 

Since Sh(F, V )N Sh(F W , V ) and C½ (V , W ) is an S 0 -open subset of C½ (V , M), 

we obtain that U¼¼ U¼ N is an S s -neighbourhood of i VÏ V M in C½ (V , M). 

Moreover, ³V coincides with ³W ,V on func(F W , V ). Therefore is also a section of 

³V defined on U¼¼ . Then by (2) of Theorem 4.2 ³V is S r,s -open. 

Emphasize that in this theorem W is an arbitrary open neighbourhood of V . 

10. Openness of Sh(F W , V) in Sh(F): Regular case 

Let z be a regular point of F, i.e. F(z)0. In this section we present necessary 

and sufficient conditions for W r,r -openness of local shift map ³V , where V is arbitrary 

small D-neighbourhood of z, see Theorem 10.2. As a consequence we will obtain 

the following 

Theorem 10.1. Suppose that z is a regular point of F having one of the following 

properties: 

(a) z is non-periodic and non-recurrent; 

(b) z is periodic and the germ at z of its first recurrence map RÏ (B, z)(B, z) is 

periodic; 

(c) z is periodic and the tangent map T z RÏ T z B T z B has at least one eigen value 

such that 1. 

Then for any sufficiently small connected D-neighbourhood V of z the corresponding 

shift map ³V of F is W r,r -open for all r 1. Moreover, if z satisfies either (a) or (b), 

then ³V is W 0,0 -open as well. 

The proof will be given in §10.8. 

Since z is a regular point of F, there exist 0, a neighbourhood W¼ of z, and 

a diffeomorphism 

Ï W¼Ên 1¢ ( 4, 4) 

such that in the coordinates (y, s) on W¼ induced by we have that 

F((y, s), t)(y, s t),


Fig. 3. Flow box. 

whenever s, s t ¾ ( 4, 4). We will call W¼ a 4-flow-box at z, see Fig. 3. Let also 

W 1 (Êm 1¢( ,))W¼ be the “central”-flow-box at z. For every D-submanifold 

V W denote 

(10.1) U V «¾ 

func(F, V )Ï³V («)(V )W. 

Then func(F W , V )U V 

and 

(10.2) ³V (U V )Sh(F, V )C½ (V , W ) 

is a W 0 -open neighbourhood of i V in Sh(F, V ). 

Let p 1Ï W¼ Êm 1 and p 2Ï W¼ ( 4, 4) be the standard projections. Then 

we can define two maps P 1Ï U V C½ (V ,Êm 1 ) and P 2Ï U V C½ (V , ( ,)) by 

³V 

P 1 («) p 1Æ³V («)Ï V 

(«) W Êm 1¢ ( ,) p 1Êm 1 , 

³V 

P 2 («) p 2Æ³V («)Ï V 

(«) W Êm 1¢ ( ,) 

2 p ( ,), 

for «¾ U V . Thus ³V («)(P 1 («), P 2 («)). 

Theorem 10.2. 

Ê be the zero function. 

(1) Then the map P 1 is locally constant with respect to the W 0 -topology of U V and its 

image P 1 (U V )C½ (V ,Êm 1 ) is at most countable. 

Let V W be a connected compact D-submanifold, and 0Ï V 

(2) The shift map ³VÏ func(F, V )Sh(F, V ) is W r,r -open if and only if P 1 (0) is an 

isolated point of P 1 (U V ) in C½ (V ,Êm 1 ) with respect to the W r -topology. 

Proof. (1) We need the following lemma: 

Lemma 10.3. 

(i) «(x) ¬(x)2 for all x¾ V , 

(ii) «(x) ¬(x)6 for all x¾ V , 

(iii) ¬(x) «(x)6 for all x¾ V . 

Moreover, condition (i) implies that 

(iv) P 1 (¬) P 1 («) and P 2 (¬) P 2 («)¬ 

Let «,¬¾ U V . Then one of the following conditions holds true: 

«.


Proof. Define the following three open, mutually disjoint subsets of V : 

K 1x¾ VÏ«(x) ¬(x)2, 

K 2x¾ VÏ«(x) ¬(x)6, 

K 3x¾ VÏ¬(x) «(x)6. 

Let x ¾ V and suppose that ¬(x) «(x)2, i.e. x K 1 . Since W is the central 

-flow-box of a 4-flow-box, and 

³V («)(V )³V (¬)(V )W , 

we see that ¬(x) «(x)6, i.e. x¾K 2K 3 . Thus V K 1K 2K 3 . Since V is 

connected and K i are open and disjoint, we obtain that V coincides with one of them. 

(i)µ(iv). Denote f ³V («), g³V (¬), and (x)¬(x) «(x). Then 

(10.3) g(x)F(x,¬(x))F(F(x,«(x)),¬(x) «(x))F( f (x),(x)) 

for all x ¾ V . Suppose that 2 on all of V . Recall that F(y, sÁ t)(y, s t) 

whenever s, t, s t ¾ ( 4, 4) and y¾Êm 1 . Since P 2 («),P 2 (¬), we obtain 

that P 2 («)3 

and 

(P 1 (¬), P 2 (¬))g (10.3) 

F(P 1 («), P 2 («)Á)(P 1 («), P 2 («)). 

P 2 («)¬ «. 

Hence P 1 (¬) P 1 («) and P 2 (¬) 

Corollary 10.4. Define the following equivalence relation on U V by 

«¬ if and only if « ¬2. 

Then every class is W 0 -open and by (iv) of Lemma 10.3 the map P 1 is defined on the 

equivalence classes, whence P 1 is locally constant. 

Moreover, there is a well-defined strict order on the equivalence classes of : if 

A and B are two distinct classes of then 

A B if and only if « ¬ 6 

on V for some «¾ A and ¬¾B. Hence there are at most countable many classes of 

, and therefore the image of P 1 is at most countable. 


We have only to show that the definition of “” does not depend on a 

particular choice of « and ¬. Let «¼¾ A and ¬¼¾B be another functions. Then 

«¼ ¬¼ («¼ «)(« ¬)(¬ ¬¼ ) 2 6 2 2. 

Hence by Lemma 10.3, «¼ ¬¼ 6 as well.


This corollary proves statement (1) of Theorem 10.2. 

(2) Let A 0 be the equivalence class ofcontaining the zero function 0, i.e. A 

«¾ U VÏ«2. Then it is easy to see that 

0 

A 0 func(F W , V ), whence Sh(F W , V )³W ,V (A 0 )³V (A 0 ). 

Thus we obtain the following commutative diagram 

A 0 func(F W , V ) ¸ ÃU V ¸ Ã func(F, V ) 

³W ,V³V ³V ³V 

¸ Ã³V (U V )Sh(F, V )C½ (V , W ) ¸ Ã Sh(F, V ) 

Ã 

³V (A 0 )Sh(F W , V ) 

Ã 

Ã 

p£1 

Ã im P 1 . 

p£1Ã 

P 1 (A 0 )¸ 

Ã 

Lemma 10.5. The image Sh(F W , V ) is W r -open in Sh(F, V ) for some r ¾ Æ0 

if and only if P 1 (0) P 1 (A 0 ) is an isolated point of the image im P 1 P 1 (U 

C½ 

V 

(V ,Êm 

) 

1 ) with respect to the W r -topology. 

Proof. Sufficiency. Suppose that P 1 (0) is an isolated point of im P 1 in the 

W r -topology, i.e. P 1 (0) is a W r -open subset of im P 1 . Since p£1 is Wr,r -continuous, we 

obtain that Sh(F W , V )³V (A 0 ) is W r -open in ³V (U V ) being W r -open in Sh(F, V ). 

Hence Sh(F W , V ) is W r -open in Sh(F, V ) as well. 

Necessity. Conversely, suppose that P 1 (0) is not isolated in im P 1 with the 

W r -topology. We will construct a sequence of functions ¬i such that 

(i) P 1 (¬i) P 1 (0), 

(ii) the sequence ³V (¬i) converges to i VÏ V M in the W r -topology. 

It will follow from (i) that ³V (¬i)Sh(F W , V )³V (A 0 ), and in particular ¬i A 0 

func(F W , V ) for all i¾Æ. Then from (ii) we will obtain that Sh(F W , V )³W ,V (A 0 ) is 

not a W r -neighbourhood of i V in Sh(F, V ), i.e. Sh(F W , V ) is not W r -open in Sh(F, V ). 

Since P 1 (0) is not isolated in im P 1 , there exists a sequence of classes A i distinct 

from A 0 such that their images P 1 (A i ) converge to P 1 (A 0 ) in the W r -topology. 

A i 

It is easy to see that for every i ¾Æ there exists a function ¬i ¾ such that 

P 2 (¬i)(x, s) P 2 (0)(x, s)s. Indeed, take any ¬¼i ¾ A i and set ¬i ¬¼i P 2 (¬¼i ). 

We claim that the sequence ¬ii¾Æ satisfies conditions (i) and (ii). Property (i) 

holds since P 1 (¬i) P 1 (A i ) P 1 (A 0 ) for all i¾Æ. 

Moreover, we have that P 2 (¬i) P 2 (0) p 2Æ i V and P 1 (¬i) C½ (V ,Êm 1 ) 

converges to P 1 (0) p 1Æi V in W r -topology. Hence³V (¬i) converges to i V with respect 

to the W r -topology. This proves (ii). 

Lemma 10.6. The map ³W ,V is W r,r -open for all r 0.



Notice that 

func(F W , V )«¾ C½ (V ,Ê)Ï«(y, s), s«(y, s), 

and 

Hence 

³W ,V («)(y, s)(y,«(y, s)s). 

Sh(F W , V )f ¾ C½ (V , W )Ï p 1Æ f (y, s) y, p 2Æ f. 

1 

and the inverse map ³ 

W ,VÏ Sh(F W , V )func(F W , V ) is given by 

1 

³ s) W ,V 2Æ ( f )(y, p f (y, s) s. 

Evidently, this map is W r,r -continuous for all r ¾Æ0, whence ³W ,V 

is W r,r -open. 

Now statement (2) of Theorem 10.2 follows from Lemmas 10.5, 10.6, and statement 

(B) of Theorem 9.1. Theorem 10.2 is completed. 

10.7. Periodic case. Let z be a periodic point of F. We will show that for a 

sufficiently small flow-box at z the images of -classes defined in Corollary 10.4 can 

be described in terms of that first recurrence map of orbit o z of z only, see (10.5). 

Let be the period of z, B be a codimension one open disk which transversally 

intersects the orbit o z at z, and RÏ (B, z)(B, z) be the first recurrence map. Let also 

5 and Ï W¼Êm 1¢ ( 4, 4) be a 4-flow-box at z. Decreasing W¼ we can 

assume, in addition, that 

(10.4) o z W¼ F(z¢ ( 4, 4)), 

B W , and B is transversal to all orbits of W¼ , so that the restriction p1BÏ W¼ 

BÊm 1 is a diffeomorphism. Put 

d (p 1B) 

1Æ p 1Ï V p 1Êm 1 (p 1B) 1 B. 

Then d preserves the first coordinate. 

Let W W¼ be the central -flow-box, V W be a connected D-neighbourhood 

of z, and A be the-class of U V . Then it follows from (10.4) that there exists a unique 

k¾ such that «(z) k for every «¾ A, and 

(10.5) P 1 (A) p 1Æ kÆ R dÏ d Rk V B B p 1Êm 1 

.


10.8. Proof of Theorem 10.1. Due to Theorem 10.2 it suffices to find a 

4-flow-box neighbourhood W¼ of z such that for every connected D-neighbourhood 

V of z, contained in the central -flow-box neighbourhood W , the image P 1 (0) is an 

isolated point of P 1 (U V ) in the W r -topology, where r 0 in the cases (a) and (b), and 

r 1 in the case (c). 

(a) Suppose that z is non-recurrent. Then there exist 0, and a 4-flow-box 

neighbourhood W¼ of z such that F(z, t)¾W¼ iff t4. Let W W¼ be the central 

-flow-box and V W be a connected D-neighbourhood. Then it follows from 

Lemma 10.3 that there exists only one equivalence class of . Hence im P 1 P 1 (0), 

and thus P 1 (0) is isolated in im P 1 in any of W r -topologies. 

Suppose that z is periodic. Let B be a codimension one open disk which transversally 

intersects the orbit o z at z, RÏ (B, z)(B, z) be the first recurrence map, and 

W¼ , W , and V be such as in §10.7. 

(b) If the germ of R at z is periodic of some period q, then we can assume that 

there exists an open neighbourhood E B of z such that R(E) E and d(V ) E. 

Then it follows from (10.5) that the image of P 1 is finite and consists of q points. 

Hence P 1 (0) is isolated in im P 1 in any of W r -topologies. 

(c) Suppose that the tangent map T z RÏ T z B T z B has at least one non-zero 

eigen vectorÚ¾T z B 

T z V with eigen valuesuch that1. Then T z R k (Ú)kÚ. 

Let T z p 1Ï T z B T 0Êm 1 be the tangent map of p 1 at z and uT z p 1 (Ú). Then for 

every class A corresponding to some k¾ we have that 

T z (P 1 (A)) (10.5) 

T z (p 1Æ kÆ R d)(Ú)k u. 

Since1, we see that P(0) is isolated in im P 1 in any of W r -topologies for r 1. 

11. Openness of Sh(F W , V) in Sh(F): Singular case 

Let W M be an open subset and V W be a D-submanifold. 

to (10.1) put 

Similarly 

(11.1) U V «¾ 

func(F, V )Ï³V («)(V )W. 

Then func(F W , V )U V 

and 

(11.2) ³V (U V )Sh(F, V )C½ (V , W ) 

is a W 0 -open neighbourhood of i V in Sh(F, V ). We will now present a sufficient condition 

which guarantees that func(F W , V )U V . Due to (11.2) this will imply that 

(11.3) Sh(F W , V ) def 

³V (func(F, V ))³V (U V )Sh(F, V )C½ (V , W )


a) b) 

Fig. 4. 

is W 0 -open in Sh(F, V ). In particular, we will prove (11.3) for vector fields of types 

(Z), (L), (H) and their products. 


Let V W be a subset. We will say that a pair (W , V ) has 

proper boundary intersection property (p.b.i.) if for any x ¾ V , a ¾Ê, and T 1 

such that 

• F(x,a)¾W for ¾ [0, 1)T, but 

• F(x, a)¾Fr(W )WÒ W , 

there exists ¼¾ (1, T ) such that F(x,¼ a)W , see Fig. 4 a). 

Roughly speaking, if the orbit o x of any x¾ V leaves W for a certain amount of 

time and comes back into W , then during this time it must leave the closure W . In 

Fig. 4 b) the pair (W , V ) does not satisfy p.b.i. The following lemma is the crucial 

implication of p.b.i.: 

Lemma 11.2. Let z¾W , F(z)0, and V W be a connected D-neighbourhood 

of z such that (W , V ) has p.b.i. Then func(F W , V )U V . 

Proof. Let « ¾ U V , so F(x,«(x)) ¾ W for all x ¾ W . We have to show that 

«¾ func(F W , V ). It suffices to verify that the following subset of V : 

K«x¾ VÏ F(x, s«(x))¾W for all s¾I [0, 1] 

coincides with all of V . Notice that z ¾ K«. Moreover, since W is open in M, it 

follows that K« is open in V . Therefore it suffices to show that VÒ K« is open in V 

as well. Since V is connected, we will get that K« V . 

Let x¾ VÒ K«. Then x F(x, 0¡«(x)) and F(x, 1¡«(x)) belong to W and there 

exists 0 ¾ (0, 1) such that F(x,«(x))¾W for all ¾ [0,0) while F(x,0«(x))¾ 

Fr(W ). Now it follows from p.b.i. for the pair (W , V ) that there exists¼¾ (0, 1) such 

that F(x,¼«(x))W . 

Then there is an open neighbourhood V x of x in V such that 

F(y,¼«(y))W


for all y¾V x . Hence V x 

VÒ K«, and thus VÒ K« is open in V . 

The following simple lemma is left for the reader. 

Lemma 11.3. (1) If (W , V ) has p.b.i., then for every subset V¼ V the pair 

(W , V¼ ) also has that property. 

(2) Let W¼ be an open neighbourhood of W . Then (W , V ) has property p.b.i. with 

respect to F if and only if it has this property with respect to the restriction FW¼. In 

other words, p.b.i. is determined by the behavior of F on arbitrary small neighbourhood 

of W only. 

(3) For every x¾ W denote by x the connected component of o x W containing x. 

Suppose that Fr(W ) is a smooth submanifold of M and for every x ¾ V its orbit o x 

is transversal to Fr(W ) at each y ¾x Fr(W ) whenever such a point exists. Then 

(W , V ) has p.b.i. 

11.4. Isolating blocks. Statement (3) of Lemma 11.3 is relevant with one of the 

principal results of [4]. An open subset U M is an isolating neighbourhood for the 

flow F if o x ê 

U for all x ¾ Fr(U)U Ò U. A closed F-invariant subset X M 

is isolated, if X is the maximal invariant subset of some isolating neighbourhood U 

of X. 

Let W be a compact D-submanifold of M and ÛW Ò Int W be its boundary. 

Denote 

Ûx¾ÛÏ0 with F(x¢ ( , 0)) W û, 

Û x¾ÛÏ0 with F(x¢ (0,)) W û, 

x¾ÛÏ F is tangent to Û. 

Then W is called an isolating block for F if ÛÛ and is a smooth submanifold 

of Û with codimension one. Thus Û and Û are submanifolds (possibly 

with corners if so is W ) ofÛwith common boundary. It follows that the interior of 

an isolating block is an isolating neighbourhood. Moreover, from (3) of Lemma 11.3 

we obtain 

Lemma 11.5. If W is an isolating block, then for any subset V Int W the pair 

(Int W , V ) has p.b.i. 

Theorem 11.6 ([4]). 

Let X M be a closed isolated F invariant subset and 

U X be its isolating neighbourhood. Then there exists an isolating block W such 

that X Int W U. 

This result was established for the case when W is a manifold with boundary, but 

if M is F-invariant and X Mû, then the proof easily extends to D-submanifolds.


Corollary 11.7. 

Suppose z is an isolated singular point of F. Then there exist a 

base ¬W««¾A of open neighbourhoods of 0¾Êk such that for every W ¾¬ and 

any subset V W the pair (W , V ) has p.b.i. 

11.8. Semi-invariant sets. We say that W is negatively (positively) invariant with 

respect to F if for every x ¾ W and t 0 (t 0) such that (x, t)¾dom(F) we have 

F(x, t)¾ W . In this case we will also say that W is semi-invariant. 

Lemma 11.9. If W is semi-invariant with respect to F, then for any subset V 

W the pair (W , V ) has p.b.i. 

Proof. Notice that if x, F(x, T a)¾W for some a¾Ê and T 1, then it follows 

from semi-invariance of W that F(x,a)¾W for all ¾ [0, T ]. Hence there are no 

x¾ V , a¾Ê and T 1 satisfying assumptions of Definition 11.1. Therefore (W , V ) 

has p.b.i. 

11.10. Product of flows. For i 1, 2 let M i be a manifold, F i be a vector field 

on M i , W i M i be an open subset, and V i W i be a subset. Denote MM 1¢M 2 

and W W 1¢ W 2 . Consider the product of these vector fields F(x, y)(F 1 (x), F 2 (y)) 

on M. It generates a local flow F(x, y, t)(F 1 (x, t), F 2 (y, t)). 

Lemma 11.11. Suppose that (W i , V i ) has p.b.i. with respect to F i , (i 1, 2), and 

let V V 1¢V 2 be a subset. Then (W , V ) has p.b.i. with respect to F. 


Let x (x 1 , x 2 )¾V , a¾Ê, and T 1 be such that 

F(x,a)(F 1 (x 1 ,a), F 2 (x 2 ,a))¾W , ¾ [0, 1)T, 

F(x, a)(F 1 (x 1 , a), F 2 (x 2 , a))¾Fr(W )(Fr(W 1 )¢ W 2 )(W 1¢ Fr(W 2 )). 

In particular, F i (x i , a)¾Fr(W i ) for at least one index i 1, 2. For definiteness assume 

that F 1 (x 1 , a)¾Fr(W 1 ). Since x 1 ¾ V 1 , it follows from p.b.i. for (W 1 , V 1 ) that there 

exists ¼ ¾ (1, T ) such that F 1 (x 1 ,¼ a) W 1 . Then F(x,¼ a) W as well. Hence 

(W , V ) has p.b.i. with respect to F. 

Corollary 11.12. Let F be a product of vector fields of types (L) or (H) on Êm . 

Then there exist a base ¬W««¾A of open neighbourhoods of 0¾Êk such that for 

every W ¾¬ and any subset V W the pair (W , V ) has p.b.i. In particular, if V is 

a connected D-neighbourhood of 0, then func(F W , V )U V and thus by (11.2) 

is W 0 -open in Sh(F, V ). 

Sh(F W , V )Sh(F, V )C½ (V , W )


Proof. Due to Lemma 11.11 it suffices to assume that F is of type (L) or (H). 

If F is of type (HE), then 0 has an arbitrary small invariant neighbourhoods W and 

by Lemma 11.9 (W , V ) has p.b.i. for any subset V W . If F is of type (HS) then 

is an isolated invariant subset of any of its small neighbourhood, whence 

0 Ê2 

existence of W follows from Theorem 11.6, though it can easily be constructed without 

referring to this theorem. 

It remains to consider the case when F(x) Bx, where B is a Jordan cell corresponding 

either to some eigen value a¾Ê or to the pair of complex conjugate a¦ ib, 

(a, b¾Ê). 

1) If a 0 (a 0), then, e.g. [26], 0 has arbitrary small positively (negatively) 

F-invariant neighbourhoods W . Moreover, if B 0 b 

, then 0¾Ê2 has arbitrary 

b 0 

small F-invariant neighbourhoods W . Then again by Lemma 11.9 such neighbourhoods 

have desired properties. 

2) Suppose that B 

¡¡¡ 

0 1 

0 1 

0 

given by the following formulas: 

F 1 (x, t)x 1x 2 t 

is nilpotent. Then the coordinate functions of F are 

t 2 

t 

x 3 x 

2!¡¡¡ m 1 

m 

(m 1)! , F i 

F i 1 

, 

t 

for i 2,, m. In particular, for each x ¾Êm these functions are polynomials in t 

of degree m 1. Hence for every Ær 0 there exists T 0 such that if x 0Ær and 

x 0 is not a fixed point of F, then F(x, t) strictly monotone increases when increases 

t T (decreases t T ). 

For every r 0 let V Êm r be the closed m-disk of radius r centered at the 

origin and W r Int(V r ) be its interior. Since F(0, t)0 for all t¾Ê, it follows that 

for every R 0 there exists rÆr such that F(V r¢ [ T , T ])W R , i.e.F(x, t) R 

for xr and tT . We claim that the pair (W R , V r ) has p.b.i. 

Suppose that for some x¾ V r and a¾Ê we have F(x,a)¾W R for¾ [0, 1) and 

F(x, a)¾Fr(W R )V 

R 

, i.e.F(x,a) R andF(x, a) R. ThenaT , whence 

F(x, t) strictly monotone increases when increases t. In particular, F(x,a) R 

for all 1, i.e. F(x,a)V R W R . 

3) Suppose that ib, (b0), is purely imagine but m 2k 4 for some k 2. 

Then regarding Êm as k we have that in complex coordinates A 

¡¡¡ 

ib 1 

ib 1 

ib . 

Hence the coordinate functions of F are given by formulae similar to the case 2). Denote 

z (z 1 ,, z k ) and p(z, t)z 1 z 2 t z 3 t 22!¡¡¡z k t k 1(k 1)!. Then 

F 1 (z, t)e ib p(z), F i e ibF i 1 

, i 2,, k. 

t


Since e ib1, it follows that F(x, t) satisfies monotonicity conditions analogous to 

the case 2). Then by the similar arguments we obtain that for every R 0 there exists 

r 0 such that the pair (W R , V r ) has property p.b.i. 

12. Proof of Theorem 3.5 

Let F be a vector field of class F(M). It follows from results of [18] that Sh(F) 

E id (F) k , where k 1 if (L)¼(HS)¼ 

and k 0 otherwise. 

Therefore we should prove that the map³Ï func(F)Sh(F) is a homeomorphism 

with respect to S½ -topologies. Actually we want to apply Theorem 8.2. 

Claim 12.1. Suppose that F belongs to class F(M). 

(1) If z¾, then for any sufficiently small connected compact D-neighbourhood V of 

z the shift map ³VÏ func(F, V )Sh(F, V ) is 

(L2)¼ 

W½,½ -open. Moreover, if 

, then ³V is even W 

z¾(L1)¼ 

r,r2 -open for all r 0. 

(2) If z is a regular point of F, then for any sufficiently small connected compact D- 

neighbourhood V of z the shift mapping ³VÏ func(F, V )Sh(F, V ) is W r,r -open for 

all r 1. 

Proof. By (C) Theorem 9.1 it suffices to find a neighbourhood W of z such that 

for every connected D-neighbourhood V W of z 

(i) the shift map³W ,VÏ func(F W , V )Sh(F W , V ) is W r,s -open for the corresponding 

values r, s, and 

(ii) its image Sh(F W , V ) is W k -open in Sh(F), where k 0 in the cases (1), and k 1 

in the case (2). 

(1) Let z¾F. Then by definition of class F(M) there exists a neighbourhood 

W of z on which in some local coordinates either 

(a) 

G n each of which belongs 

F is a product of finitely many vector fields G 1 ,, 

to one of the types (Z), (L), or (H), or 

(b) F belongs to one of the types (L1)¼ , (L2) 

¼ , (L3) 

¼ , (H) 

¼ 

and z is an isolated singular 

point of F. 

Decreasing W we can also assume that Sh(F W , V ) is W 0 -open in Sh(F), i.e. condition 

(ii) is satisfied. In the case (a) this follows from Corollary 11.12, while in the 

case (b) from Corollary 11.7. 

Then (i) directly follows from Lemmas 7.1–7.3. 

(2) Suppose that z is a regular point for F. Since F belongs to F(M), it follows 

that the assumptions of Theorem 10.1 are satisfied, whence for any sufficiently small 

D-neighbourhood of z the shift map ³V is W r,r -open for all r 1. 

For every z ¾ M let V z be a neighbourhood guaranteed by Claim 12.1. By assumption 

of Theorem 3.5 the set Ò ((L1)¼(L2)¼ 

) is compact.


Therefore using paracompactness of M we can find a locally finite cover V ii¾¿ 

of M by compact connected D-submanifolds and a finite subset ¿¼¿ such that the 

map ³V i 

is W½,½ -open for i¾¿¼ and W r,r2 -open for all r 0 whenever i¾¿Ò¿¼ . 

Hence by “fragmentation” Theorem 8.2 ³ is a local homeomorphism with respect to 

S½ -topologies. All other statements concerning homotopy types of Did (F) r and E id (F) r 

follow by the arguments of the proof of [13, §9]. Theorem 3.5 is completed. 

ACKNOWLEDGEMENTS. I would like to thank V.V. Sharko, A. Prishlyak, 

I. Vlasenko, and I. Yurchuk for useful discussions. I am especially grateful to E. Polulyakh 

for very helpful conversations and pointing out to the irrational flow on T 2 which lead 

me to discovering mistakes of [13]. 

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Topology dept. 

Institute of Mathematics of NAS of Ukraine 

Tereshchenkivs’ka st. 3, Kyiv, 01601 

Ukraine 

e-mail: maks@imath.kiev.ua

Title Local inverses of shift maps along orbits of ... - Osaka University

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