Reversing orientation homeomorphisms of surfaces

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. is a mirror symmetry with respect to some line. The obtained results hold in fact for a larger class of maps with isolated singularities from connected compact orientable surfaces to the real line and the circle.


Introduction
The present paper describes several foliated and homotopy variants of a "rigidity" property for reversing orientation linear motions of the plane claiming that every such motion has order 2. Though it is motivated by study of deformations of smooth functions on surfaces (and we prove the corresponding statements), the obtained results seem to have an independent interest.
Let D n = {r, s | r n = s 2 = 1, rs = sr −1 } be the dihedral group, i.e. group of symmetries of a right n-polygon. Then each "reversing orientation" element is written as r k s for some k and has order 2: (r k s) 2 = r k sr k s = r k r −k ss = 1.
Nevertheless, counterparts of the above rigidity effects for homeomorphisms can still be obtained on a "homotopy" level.
For instance, let H(S 1 ) be the group of all homeomorphisms of the circle S 1 and H + (S 1 ), (resp. H − (S 1 )), be the subgroup (resp. subset) consisting of homeomorphisms preserving (resp. reversing) orientation. Endow these spaces with compact open topologies. Notice that we have a natural inclusion O(2) ⊂ H(S 1 ), which consists of two inclusions SO − (2) ⊂ H − (S 1 ) and SO(2) ⊂ H + (S 1 ) between the corresponding path components. It is well known and is easy to see SO − (2) (resp. SO (2)) is a strong deformation retract of H − (S 1 ) (resp. H + (S 1 )). This implies that the map sq : H − (S 1 ) → H + (S 1 ) defined by sq(h) = h 2 is null homotopic.
The aim of the present paper is to prove a parametric variant of the above "rigidity" statements for self-homeomorphisms of open subsets of topological products X × S 1 preserving first coordinate (Theorem 6.2). That result will be applied to diffeomorphisms preserving a Morse function on an orientable surface and reversing certain regular components of some of its level-sets (Theorems 3.3, 3.5, and 3.6).

2.2.
Homogeneous polynomials on R 2 without multiple factors. It is well known and is easy to prove that every homogeneous polynomial f : R 2 → R is a product of finitely many linear and irreducible over R Q j , where L i (x, y) = a i x + b i y and f = L1 · · · L l Q1 · · · Qq f = Q1 · · · Qq f = L1Q1 · · · Qq f = L1L2 l ≥ 2 l = 0, q ≥ 1 l = 1, q ≥ 1 l = 2, q = 0

2.3.
Several constructions associated with f ∈ F(M, P ). In what follows we will assume that f ∈ F(M, P ). Let Σ f be the set of critical points of f . Then condition (A2) implies that each z ∈ Σ f is isolated. In the case when P = S 1 one can also say about local extremes of f , and even about local minimums or maximums if we fix an orientation of S 1 .
A connected component K of a level-set f −1 (c), c ∈ P , will be called a leaf (of f ). We will call K regular if it contains no critical points. Otherwise, it will be called critical .
For ε > 0 let N ε be the connected component of f −1 [c−ε, c+ε] containing K. Then N ε will be called an f -regular neighborhood of K if ε is so small that N ε \ K contains no critical points of f and no boundary components of ∂M . In other words, each critilal leaf of f is additionally partitioned by critical points. We will call Ξ f the singluar foliation of f .

2.3.4.
Hamiltonian like flows of f ∈ F(M, P ). Suppose M is orientable. A smooth vector field F on M will be called Hamiltonian-like for f if the following conditions hold true.
(a) F (z) = 0 if and only if z is a critical point of f .
Moreover, in this case the homotopy H : M × [0; 1] → M given by H(x, t) = F(x, tα(x)) is an isotopy between H 0 = id M and H 1 = h in S(f ).

Main results
Let M be a compact surface, h : M → M a homeomorphism, and γ ⊂ M a submanifold of M . Then γ is h-invariant, whenever h(γ) = γ.
We will be interesting in the structure of diffeomorphisms preserving f ∈ F(M, P ) and reversing orientations of some regular leaves of f . The following easy lemma can be proved similarly to [6,Lemma 3.5].
Lemma 3.1. Let M be a connected orientable surface and h ∈ S(f ) be such that every regular leaf of f is h-invariant. Then every critical leaf of f in also h-invariant and the following conditions are equivalent: (1) some regular leaf of f is h + -invariant; (2) all regular leaves of f are h + -invariant; (3) h preserves orientation of M . Let h ∈ ∆ − (f ). Then by Lemma 3.1 every critical leaf K of f is h −invariant, however, h may interchange critical points of f in K and the leaves of Ξ f contained in K (i.e. connected components of K \ Σ f ). Notice also that in general ∆ − (f ) can be empty.
The following theorem can be regarded as a homotopical and foliated variant of the above rigidity property for diffeomorphisms preserving f .  2), and that V is h − -invariant. Then there exists g ∈ S(f ) which coincide with h on some neighborhood of V and such that g 2 ∈ S id (f ).
Emphasize that Theorem 3.5 does not claim that g ∈ ∆ − (f ), though g reverses orientation of V and its square belong to S id (f ).
General result. Theorems 3.3 and 3.5 are consequences of the following Theorem 3.6 below. Let M be a connected compact (not necessarily orientable) surface, f ∈ F(M, P ), and h ∈ S(f ). Let also • A be the union of all h − -invariant regular leaves of f , • K 1 , . . . , K k be all the critical leaves of f such that A ∩ K i = ∅; Evidently, Z is an f -adapted subsurface of M and each of its connected components intersects A.
Lemma 3.5.1. Suppose Z is orientable. Let also γ be a boundary component of Z. Consider the following conditions: Then (2)⇒(4) Suppose h(γ) = γ. Let Z be a connected component of Z containing γ. Then by the construction Z must intersect A, whence h changes orientation of some regular leaves in Z . Since Z is also orientable, we get from Lemma 3.1 that h also changes orientation of γ. This means that γ ⊂ A, and therefore there exists an open neighborhood W ⊂ A of γ consisting of regular leaves of f . In particular, the regular leaves in W ∩Int U must be contained in A which contradicts to the assumption that Thus h interchanges boundary components of Z belonging to the interior of M . We will introduce the following property on Z: This condition means that there is a bijection between boundary components of ∂Z ∩ Int M and connected components of M \ Z, whence by Lemma 3.5.1 there will be no h-invariant connected components of M \ Z.
Theorem 3.6. If Z is non-empty, orientable and has property (B), then there exists g ∈ S(f ) such that g = h on Z and g 2 ∈ S id (f ).
Proof of Theorem 3.3. Suppose M is orientable and h ∈ ∆ − (f ). Then in the notation of Theorem 3.6, Z = M , and by that theorem there exists g ∈ S(f ) such that g = h on Z and g 2 ∈ S id (f ). This means that h = g and h 2 ∈ S id (f ).
Proof of Theorem 3.5. The assumption that every regular leaf of f in Int M separates M implies condition (B).

3.7.
Structure of the paper. In Section 4 we discuss a notion of a shift map along orbits of a flow which was studied in a series of papers by the second author, and extend several results to continuous flows. Section 5 devoted to reversing orientation families of homeomorphisms of the circle. In Section 6 we study flows without fixed point, and in Section 7 recall several results about passing from a flow on the plane to the flow written in polar coordinates. Section 8 introduces a certain subsurfaces of a surface M associated with a map f ∈ F(M, P ) and called chipped cylinders. We prove Theorem 8.3 describing behaviour of diffeomorphisms reversing regular leaves of f contained in those chipped cylinders. In Section 9 we prove Lemma 9.1 allowing to change f -preserving diffeomorphisms so that its finite power will be isotopic to the identity by f -preserving isotopy. Finally, in sections 9 and 10 we prove Theorem 3.6.

Shifts along orbits of flows
In this section we extend several results obtained in [4,8] for smooth flows to a continuous situation. Let X be a topological space.
It is well known that a C r , 1 ≤ r ≤ ∞, vector field F of a smooth compact manifold X tangent to ∂X always generates a flow F.
Assume further that F is a flow on a topological space X. Let also V ⊂ X be a subset. Say that a continuous map h : Notice that in general α x is not unique and does not continuously depend on x.
Conversely, let α : V → R be a continuous function such that its graph For a global flow any continuous α : V → R satisfies that condition. Then one can define the following map We will call F α a shift along orbits of F by the function α, while α will be called a shift function for F α . Notice that F α preserves orbits of F on V , and in general is not a homeomorphism.
In other words, F is locally conjugated to the flow G : (Y ×R)×R → Y ×R defined by G(y, s, t) = (y, s + t), since the identity (4.1) can be written as It is well known that each C 2 flow (generated by some C 1 vector field) on a smooth manifold admits flow-box charts at each non-fixed point.
Let also ε and V be as in Definition 4.1, and V = F τ (V ) be a neighborhood of y. Then for any The following lemma shows that for flows admitting flow box charts (e.g. for smooth flows) every orbit preserving map admits a shift function near each non-fixed point. Moreover, such a function is locally determined by its value at that point.
If in addition X is a manifold of class C r , (1 ≤ r ≤ ∞), F and h are C r , and φ is a C r embedding, then α is C r as well.
Proof. The proof almost literally repeats the arguments of [8, Lemma 6.2] proved for smooth flows and based on existence of flow box charts. For completeness we present a short proof for continuous situation.
Corollary 4.5. Suppose U ⊂ X is an open connected subset such that every x ∈ U is non-fixed and admits a flow-box. Let also α, α : U → R be two continuous functions such that F α = F α on U . If α(x) = α (x) at some x ∈ U (this holds e.g. if F has at least one non-periodic point in U ), then α = α on U .

this set is open, whence
Corollary 4.6. Let F : X × R → X be a continuous flow, U ⊂ X be an open subset such that every point x ∈ U is non-fixed and non-periodic and admits a flow box chart, and h : U → X be an orbit preserving map. Then there exists a unique continuous function α : If in addition X is a manifold of class C r , (0 ≤ r ≤ ∞), F is C r and admits C r flow box charts, and h is C r , then α is C r as well.
Proof. Since every point x ∈ U is non-fixed and non-periodic, there exists a unique number α(x) such that h(x) = F(x, α(x)). Moreover, since F admits flow box chart at x, it follows from Lemma 4.4 that the correspondence x → α(x) is a continuous function α : U → R, which is also C r under the corresponding smoothness assumptions on X, F, φ, and h.
The following statement extends some results established in [8] for smooth flows to continuous flows having flow box charts at each non-fixed point on arbitrary topological spaces. (c.f. [8, Lemmas 6.1-6.3]) Let F : X × R → X be a flow, p :X → X a covering map, and ξ :X →X a covering transformation, i.e. a homeomorphism such that p • ξ = p. Then the following statements hold.
(1) F lifts to a unique flowF : (2) For each continuous function α : (4) For a continuous function β :X → R and a point z ∈X consider the following ("global" and "point") conditions: Then we have the following diagram of implications: , whence all conditions in the right square of (4.3) are equivalent. (iii) IfX is path connected, andF has no fixed points and admits flow box charts at each point ofX, then (g2)&(p1) ⇒ (g1). Proof.
(1) Consider the following homotopy (with "open ends") and letF : 0). HenceF extends to a unique liftingF :X × R →X of G. One easily checks that this lifting is a flow onX.
(2) Let z ∈X and t = α • p(z). Then (3) Notice that the mapF : ThusF andF are two liftings of F which coincide at t = 0, and thereforẽ F =F by uniqueness of liftings. Therefore for any continuous functioñ α : (4) The implication in Diagram (4.3) are trivial. Assume thatX (and therefore X) are path connected.
(i) If z is non-fixed and non-periodic, thenF(z, a) =F(z, b) implies that a = b for any a, b. In particular, this hold for a = β(z) and b = β • ξ(z).
(ii) Suppose thatF β is a lifting of some continuous map h : X → X.
at some point z ∈X, then these liftings must coincide on all ofX, which means condition (g3).
Let p :X → X be a regular covering map with path connectedX and X, G be the group of covering transformation, F : X × R → X be a flow on X, andF :X × R →X be its lifting as in Lemma 4.7(1).
Again ifX, X, p, h, F and its flow box charts are C r , (1 ≤ r ≤ ∞), then α is also C r .
Proof. Due to assumptions onF we get from Corollary 4.6 that there exists a unique continuous function β :X → R such thath =F β . Let ξ ∈ G. Sinceh commutes with ξ, i.e. condition (g3) of Lemma 4.7 holds, we obtain the following implications: meaning that β • ξ = β. Thus β is invariant with respect to all ξ ∈ G, and therefore it induces a unique function α : X → R such that β = α • p. Since p is a local homeomorphism, it follows that α is continuous. Moreover, by Statements about smoothness of α follows from the corresponding smoothness parts of used lemmas. We leave the details for the reader.

Self maps of the circle
Let S 1 = {z ∈ C | |z| = 1} be the unit circle in the complex plane, p : R → S 1 be the universal covering map defined by p(s) = e 2πis and ξ(s) = s + 1 be a diffeomorphism of R generating the group of covering slices Z.
Denote by C k (S 1 , S 1 ), k ∈ Z, the set of all continuous maps h : S 1 → S 1 of degree k, i.e. maps homotopic to the map z → z k . Then {C k (S 1 , S 1 )} k∈Z is a collection of all path components of C (S 1 , S 1 ) with respect to the compact open topology.
For a map h : X → X it will be convenient to denote the composition h • · · · • h n by h n for n ∈ N. A point x ∈ X is fixed for h, whenever h(x) = x.
Lemma 5.1. Let h : S 1 → S 1 be a continuous map andh 0 : R → R be any lifting of h with respect to p, i.e. a continuous map making commutative the following diagram: For a ∈ Z define the maph a : R → R byh a = ξ a •h 0 , that ish a (s) =h 0 (s) + a. Then the following conditions are equivalent: Moreover, let k be the degree of h. Then (i) h has at least |k − 1| fixed points; (ii) {h ak } a∈Z is the collection of all possible liftings of h; ( Hence one can put g :=h 2 0 =h 2 a and this map does not depend on a ∈ Z. Let alsoÃ be the set of fixed points of g. We have to show thatÃ = p −1 (A).
Conversely, let z ∈ A and s ∈ R be such that z = p(s). Then there exists a unique liftingh a of h such thath a (s) = s. But then g(s) =h 2 a (s) = s, whence s ∈Ã.
The following example shows that the effect described in the statement (iv) of Lemma 5.1 includes the rigidity property of reflections of the circle mentioned in the introduction.
Example 5.2. Let h(z) = ze −2πφ e 2πφ =ze 2φ be a reflection of the complex plane with respect to the line passing though the origin and constituting an angle φ with the positive direction of x-axis. Then h is an involution preserving the unit circle and the restriction h| S 1 : S 1 → S 1 is a map of degree −1 belonging to SO − (2). Moreover, each its lifting h a : R → R is given byh a (s) = a + φ − s. But thenh 2 a = id R 2 and does not depend on a. Moreover, the set of fixed points ofh 2 a is R which coincides with p −1 (S 1 ), where S 1 is the set of fixed points of h 2 = id S 1 .
Another interpretation of the above results can be given in terms of shift functions.
Corollary 5.3. Let F : S 1 × R → S 1 be a flow on the circle S 1 having no fixed points, so S 1 is a unique periodic orbit of F of some period θ.
(1) Let h : S 1 → S 1 be a continuous map. Then h ∈ C 1 (S 1 , S 1 ) if and only if there exists a continuous function α : S 1 → R such that h = F α . Such a function is not unique and is determined up to a constant summand nθ for n ∈ Z. If F and h are C r , (0 ≤ r ≤ ∞), then so is α.
Proof. LetF : R × R → R be a unique lifting of F with respect to the covering p. Then R is a unique non-periodic orbit ofF.
(1) If α : S 1 → R is any continuous function, then the map F α is homotopic to the identity by the homotopy {F tα } t∈[0;1] , whence F α has degree 1 (the same as id S 1 ).
Conversely, let h ∈ C 1 (S 1 , S 1 ) and leth be any lifting of h. Then by Corollary 4.6 there exists a unique continuous function β : R → R such that h =F β . Moreover, since h is a map of degree 1, we get from Lemma 5.1(b), thath commutes with ξ, i.e. condition (g3) of Lemma 4.7 holds. Sincẽ h = F β is a lifting of h, Lemma 4.7(ii) implies that condition (g1) of Lemma 4.7 also holds, i.e. β • ξ = β. Hence β induces a unique function α : S 1 → R such that β = α • p andh =F α•p . Therefore By Lemma 4.7(2), h is a lifting of F α . But it is also a lifting of h, whence h = F α .
Conversely, let y ∈ A, so h(y) = y, and let x ∈ R be any point with p(x) = y. Then there exists a unique liftingh of h withh(x) = x and by Lemma 5.1(iv), p −1 (A) is the set of fixed points ofh 2 . Moreover, sinceh 2 has degree 1, we get from (1) thath 2 =F α•p . Since x is a fixed point ofh 2 as well, we must have that 0 = α • p(x) = α(y).
(3) Consider two flows of R and S 1 respectively: Evidently,F is a lifting of F, and h t = F αt . Then by Lemma 4.7(2) the maph Since h 0 and h 1 are homeomorphisms (diffeomorphisms of class C r ), it follows that so areh 0 and h 1 are homeomorphisms. Therefore so are their convex linear combinatioñ h t and the induced map h t .

Flows without fixed points
Let X be a Hausdorff topological space and F : Y × R → Y a continuous flow on an open subset Y ⊂ X × S 1 satisfying the following conditions: (Φ1) the orbits of F are exactly the connected components of the intersections (x × S 1 ) ∩ Y for all x ∈ X.
(Φ2) F admits flow box charts at each point (y, s) ∈ Y ; (Φ3) the set B = {x ∈ X | x × S 1 ⊂ Y } is dense in X; (Φ4) for each x ∈ X, the intersection (x × S 1 ) ∩ Y has only finitely many connected components (being by (Φ1) orbits of F). Thus every orbit of F is either x × S 1 or some arc in x × S 1 , and, in particular, F has no fixed points. Also condition (Φ3) implies that the set of periodic orbits of F is dense in X × S 1 . Then the complement to B: consists of x ∈ X for which x × S 1 contains a non-closed orbit of F.
It will also be convenient to use the following notations for x ∈ X: Example 6.1. Let X be a smooth manifold, and G be a vector field on X × S 1 defined by G(x, s) = ∂ ∂s , so its orbits are the circles x × S 1 . Let Y ⊂ X × S 1 . Then it is well known that there exists a non-negative C ∞ function α : be the flow on X × S 1 generated by F . Then Y is the set of non-fixed points of F and the induced flow on Y satisfies conditions (Φ1) and (Φ2). Another two conditions (Φ3) and (Φ4) depend only on a choice of Y . For instance they will hold if the complement (X × S 1 ) \ Y is finite. Theorem 6.2. Suppose F satisfies (Φ1)-(Φ4). Let also h : Y → Y be a homeomorphism having the following properties.
(1) h(L x ) ⊂ x × S 1 for each x ∈ X, so h preserves the first coordinate, and in particular, leaves invariant each periodic orbit of F, thought it may interchange non-periodic orbits contained in x × S 1 .
(2) For each x ∈ B the restriction h : x × S 1 → x × S 1 has degree −1 as a self-map of a circle. Then there exists a unique continuous function α : Y → R such that (a) h 2 (x, s) = F(x, s, α(x, s)) for all (x, s) ∈ Y , in particular h 2 preserves every orbit of F; (c) if in addition X is a manifold of class C r , (0 ≤ r ≤ ∞), F is C r and admits C r flow box charts, and h is C r , then α is C r as well.
Proof. Let p : X × R → X × S 1 be the infinite cyclic covering of X × S 1 defined by p(x, s) = (x, e 2πis ) and ξ(x, s) = (x, s + 1) be a diffeomorphism of X × R generating the group Z of covering slices. In particular, p • ξ = p.
Then the orbits ofF are exactly the connected components of L x for all x ∈ X. In particular, all orbits ofF are non-closed. Evidently, for x ∈ X the following conditions are equivalent: Then the following conditions hold.
(ii) ifh 1 is another lifting of h, thenh 2 1 =h 2 ; (iii) for each x ∈ X the restrictionh : L x → L x is strictly decreasing in the sense that if h(x, s) = (x, t) for some s, t ∈ R, then s > t; (iv) for each x ∈ X the restrictionh 2 leaves invariant each orbit ofF.
(ii) Let x ∈ B, so x × S 1 ⊂ Y is an orbit of F. Then the restriction p| x×R : x×R → x×S 1 is a universal covering map, h| x×S 1 : x×S 1 → x×S 1 is a map of degree −1, andh| x×R ,h 1 | x×R : x × R → x × R are two lifting of h| x×S 1 . Hence by Lemma 5.1(iv) Sinceh is a homeomorphism, it follows that t 0 = t 1 . Suppose that t 0 < t 1 . Then there exist a > 0, and two open neighborhoods V ⊂ U of x in X such thath Due to property (Φ3), the set B is dense in X, so there exists a point y ∈ B ∩ V = ∅. Then on the one hand y × R ⊂ Y is an orbit ofF, andh : y × R → y × R reverses orientation, so if (y, t i ) =h(y, s i ), i = 0, 1, then t 0 > t 1 .
On the other hand, due to (6.1) t i ∈ (t i − a; t i + a), whence by (6.2): which gives a contradition. Hence t 0 > t 1 .
(iv) If x ∈ B, then L x = x × R is an orbit ofF and by (i) it is invariant with respect toh. Hence it is also invariant with respect toh 2 .
Suppose x ∈ X \ B. Then by property (Φ4), if L x = (x × S 1 ) ∩ Y consists of finitely many connected components I 0 , . . . , I n−1 for some n enumerated in the cyclical order along the circle x × S 1 . This implies that L x is a disjoint union of countably many open intervals I i , (i ∈ Z), being orbits of F which can be enumerated so that ξ( I k ) = I k+n and p( I k ) = I k mod n .
Sinceh is a strictly decreasing homeomorphism of L x , it follows that there exists a ∈ Z such thath( I k ) = I a−k . Hencẽ Suppose, in addition, that the Jacobi matrix J(h, 0) of h at 0 is orthogonal. Then there exists a ∈ R such that for each s ∈ R Hence in the second case (when h reverses orientation)h 2 (0, s) = (0, s), that ish is always fixed on 0 × R.
Let F andF be the local flows generated by F andF respectively. Then F t • p(ρ, φ) = p •F t (ρ, φ) whenever all parts of that identity are defined.
3) If 0 is a degenerate critical point of f , so deg f ≥ 3, then the situation is more complicated. Notice that in this caseF is zero on ∂H = R × 0, whenceF is fixed on that line. Again the formulas for F andF are highly complicated. be an open neighborhood of the origin 0 ∈ R 2 , h : U → R 2 an embedding which preserves orbits of F, and α : U \ 0 → R be a C ∞ function such that h(x) = F(x, α(x)) for all x ∈ U \ 0. Let alsoh : p −1 (U ) → H be any lifting of h as in Lemma 7.1. Then α can be defined at x so that it becomes C ∞ in U in the following cases: In what follows we will use the following notations.
(i) K 1 , . . . , K k denote all the critical leaves of f , and (iii) Let also L 1 , . . . , L l be all the connected components of M \ K; (iv) For each i = 1, . . . , l let Then there exist a finite subset Q i ⊂ {−1, 1} × S 1 , and an immersion φ i : [−1, 1] × S 1 \ Q i → N i and a C ∞ embedding η : [0, 1] → P such that the following diagram is commutative: where p 1 is the projection to the first coordinate. Notice that φ can be noninjective only at points of {−1, 1} × S 1 and this can happens only when P = S 1 , see Example 8.1 and Figure 8.2d) below.
We will call N i a chipped cylinder of f , see Figure 8.1. It will also be convenient to denote We will call N − i and N + i chipped half-cylinders of N i and f , and Int N i the interior of N i .
(v) Let also be the union of the chipped cylinder N i with f -regular neighborhoods of critical leaves of f which intersect the closure N i . We will call Z i an fregular neighborhood of N i . c) Let f : S 2 → P be a map of class F(S 2 , P ) having only two critical points z 1 and z 2 being therefore extremes of f , see Figure 8.2c). Then K = {z 1 , z 2 }, f has a unique chipped cylinder N = S 2 \ {z 1 , z 2 } and its f -regular neighborhood is all S 2 . d) Let M be either a 2-torus or Klein bottle with a hole and f : M → S 1 be a map of class F(M, S 1 ) schematically shown in Figure 8.2d). It has only one critical point z and that point is a saddle, a unique critical leaf K = K, and two chipped cylinders N 1 and N 2 . It follows from the Figure 8.2d) that N 1 intersects only one K from "both sides", in the sense that both intersections N − 1 ∩ K and N + 1 ∩ K are non-empty. e) Let f : [0, 1]×S 1 → R be a Morse function having one minimum z and one saddle point y as in Figure 8.3. Then f has two critical leaves: the point z and a critical leaf K containing y, and three chipped cylinders N 1 , N 2 , N 3 . Let R K be an f -regular neighborhood of K. Then the corresponding f -regular neighborhoods of chipped cylinders are the following ones:   (1) N − and N + are orientable manifolds. Moreover, if P = R, then N is orientable as well. However, if P = S 1 , then it is possible to construct an example of f having non-orientable chipped cylinder, see Example 8.1d).
(2) Each of the closures N − and N + intersects at most one critical leaves of f , and those intersections consist of open arcs being leaves of the singular foliation Ξ f .  Proof. 1) Let us mention that since h reverses orientation of V , it reverses orientations of all regular leaves in N . Therefore those leaves are (h 2 , +)invariant, and we should prove the same for all other leaves of Ξ f in Z.
First we introduce the following notation. If K ∩ N − = ∅, then let K − be a unique critical leaf of f intersecting N − , and let R K − be its f -regular neighborhood. Otherwise, when K ∩ N − = ∅, put K − = R K − = ∅, Define further K + and R K + in a similar way with respect to N + . Then As those four sets are invariant with respect to h, it suffices to prove that h 2 preserves leaves of Ξ f with their orientation for each of those sets.
1a) Let us show that h 2 preserves all leaves of Ξ f in N − . Since N − is an orientable manifold, one can construct a Hamiltonian like flow of f on N − . Evidently, F satisfies conditions (Φ1)-(Φ4). Moreover, since h reverses orientation of all periodic orbits of F in N + , we get from Theorem 6.2 that there exists a unique C ∞ function α : N → R such that h 2 | N − = F α and α vanishes at fixed points of h 2 on regular leaves of f in N − . In particular, each non-periodic orbit of F in N + is (h 2 , +)-invariant as well.
1b) Now let us prove that each leaf of Ξ f in K − is also (h 2 , +)-invariant. This will imply (h 2 , +)-invariantness of all leaves of Ξ f in R K − (see the proof of the implication (iii)⇒(i) in [8,Lemma 7.4]).
If K − = ∅ there is nothing to prove. If K − ∩ N − = ∅, then by Lemma 8.2(3) K − is a local extreme of f . Hence K is an element of Ξ f and its is evidently invariant with respect h, and therefore with respect to h 2 .
Let us recall a simple proof of that fact. Indeed, let v be a vertex of γ being therefore a critical point of f . Then h(v) = v, whence h 2 preserves the set of all edges incident to v. Moreover, as h 2 preserves orientation at v, it must also preserve cyclic order of edges incoming to v. But since γ (being one of those edges) is (h 2 , +)-invariant, it follows that so are all other edges incident to v. Applying the same arguments to those edges and so on, we will see that h 2 preserves all edges of K with their orientation.
The proofs for N + and R K + are similar.
2) Assume now that Z is an orientable surface and let F be any Hamiltonian like flow of f . We know from 1) that h 2 preserves all orbits of F with their orientations. 2a) We claim that there exists a unique C ∞ function α : N → R such that h 2 | N = F α and α vanishes at fixed points of h 2 on periodic orbits.
If both K − and K + are non-empty and distinct, then the restriction of F to N satisfies conditions (Φ1)-(Φ4). Moreover, as h reverses orientation of all periodic orbits of F, the statement follows from Theorem 6.2 as in 1a).
However, if K − = K + , as in Example 8.1d), the situation is slightly more complicated: N might be not of the form ([−1, 1] × S 1 ) \ Q for some finite set Q ⊂ {−1, 1} × S 1 and Theorem 6.2 is not directly applicable. Nevertheless, one can apply that theorem to each of the sets N − , N + , and Int N and construct three functions , and vanishing at fixed points of h 2 on periodic orbits. From uniqueness of such functions, we get that α − = α 0 on N − ∩ Int N and α + = α 0 on N + ∩ Int N .
A possible problem is that N intersects K − from "both sides", and therefore a priori α + and α − can differ on N − ∩ N + ∩ K − . However, N − ∩ N + ∩ K − consists of non-periodic orbits of F, and therefore foreach such orbit γ the identity Thus α − = α + on N − ∩ N + ∩ K − , and therefore those functions define a well defined C ∞ function α : N → R satisfying h 2 | Int N = F α and α vanishes at fixed points of h 2 on periodic orbits.
2b) It remains to show that α extends to a shift function for h 2 on R K − ∪ R K + and thus on all of Z. It suffices to prove that for R K − .
If K − = ∅, then R K − = ∅ and there is nothing to prove. If K − is a local extreme of f , then by Lemma 7.3 (cases (a) and (c) for non-degenerate and degenerate critical point) α can be defined at K − so that it becomes C ∞ .
In all other cases K − contains a non-periodic orbit of F. Then by the implication (ii)⇒(iv) of [8,Lemma 7.4], α extends to a C ∞ shift function for h 2 on R K − . It remains to prove the following lemma: Proof. Indeed, it is evident, that arbitrary small neighborhood of z contains a periodic orbit γ of F. Since h reverses orientations of γ, we have from Lemma 5.1(i) that h always has at least one fixed point x ∈ γ (in fact it has even two such points). Hence by Corollary 5.3(b), α(x) = 0. Then by continuity of α we should have that α(z) = 0 as well. Theorem 8.3 is completed.

Creating almost periodic diffeomorphisms
Let M be a compact orientable surface, f ∈ F(M, P ), Z be an f -adapted subsurface, h ∈ S(f ) be such that h(Z) = Z, and m ≥ 2. If h m | Z is isotopic to the identity of Z by f -preserving isotopy, then the following Lemma 9.1 gives conditions when one can change h on M \ Z so that its m-power will be f -preserving isotopic to the identity on all of M . The proof follows the line of [8, Lemma 13.1(3)] in which M is a 2-disk or a cylinder.   can be enumerated as follows: mod m for all i, j, that is h cyclically shifts columns in (9.1). Then there exists g ∈ S(f ) such that g = h on Z and g m ∈ S id (f ), that is g m = F β for some C ∞ function β : M → R.
Y i,j , j = 0, . . . , m − 1, be the union of components from the same column of (9.1). Then h(Y j ) = Y j+1 mod m . Notice that condition (3) implies that h m (Y i,j ) = Y i,j for all i, j.
We will show that the the desired diffeomorphism g ∈ S(f ) can be defined by the formula: Indeed, by definition g = h on Z. Moreover, as τ is fixed on some neighborhood of Z, it also fixed near Z ∩ Y m−1 . Therefore g = h = τ −1 • h near Z ∩ Y m−1 , and so g is a well defined C ∞ map. It remains to prove that g m = F β ∈ S id (f ) for some C ∞ function β : M → R.
Let F be a Hamiltonian flow for f . Since τ and h m are isotopic in S(f ), it follows that τ −1 • h m ∈ S id (f ). Hence by Lemma 2.3.5, τ −1 • h m = F α for some C ∞ function α : M → R.
Since S id (f ) is a normal subgroup of S(f ), it follows that as well. Therefore, again by Lemma 2.3.5, h j • τ −1 • h m−j = F α j for some C ∞ function α j : M → R.
As τ is fixed on some neighborhood of Z, it follows that for each j Then the assumption (1) that every connected component Z of Z contains a saddle point, implies that F has a non-closed orbit γ in Z . Therefore α = α j on γ. Since Z is connected, it follows from local uniqueness of shift functions for τ −1 • h m | Z (see Corollary 4.5) that α = α j near Z for all j = 0, . . . , m − 1. Hence α = α j near all of Z for all j = 0, . . . , m − 1.
We claim that g m = F β . a) Indeed, if x ∈ Z, then g m (x) = h m (x) = F α (x) = F β (x). b) Also notice that g(Y i,j ) = Y i,j+1 mod m and g(Y j ) = Y j+1 mod m . Then g m | Y j is the following composition of maps: 10. Proof of Theorem 3.6 Let M be a connected compact surface, f ∈ F(M, P ), h ∈ S(f ), A be the union of all regular leaves of f being h − -invariant and K 1 , . . . , K k be all the critical leaves of f such that A ∩ K i = ∅. For i = 1, . . . , k, let R K i be an f -regular neighborhood of K i chosen so that R K i ∩ R K j = ∅ for i = j and Assume that Z is non-empty, orientable and every connected component γ of ∂Z ∩ Int M separates M . We have to prove that there exists g ∈ S(f ) which coincide with h on Z and such that g 2 ∈ S id (f ).
Lemma 10.1. There exists a unique C ∞ function α : Z → R such that h 2 | Z = F α and α = 0 at each fixed point of h 2 on h − -invariant regular leaves of f .
Proof. Let V be a regular leaf V of f , and N a chipped cylinder of f such that V ⊂ Int N . If V is h − -invariant, then so is every other regular leaf V ⊂ Int N . This implies that Z is a union of f -regular neighborhoods Z 1 , . . . , Z l of some chipped cylinders N 1 , . . . , N l of f . By Theorem 8.3, for each i = 1, . . . , l there exists a unique C ∞ function α i : Z i → R such that h 2 | Z i = F α i and α = 0 at each fixed point of h 2 on each on h − -invariant regular leaf of f in Z i .
Notice that if Z i ∩ Z j = ∅, then every connected component W of that intersection always contains a non-periodic orbit γ of F. Therefore by uniqueness of shift functions (Corollary 4.5) we obtain that α i = α j on W .
Hence the functions {α i } i=1,...,l agree on the corresponding intersections, and therefore they define a unique C ∞ function α : Z → R such that h 2 | Z = F α . Then α = 0 at each on h − -invariant regular leaf of f in Z i .
If M = Z, then theorem is proved. Thus suppose that M = Z.
Lemma 10.2. The number of connected components M \ Z is even, and they can be enumerated by pairs of numbers: for some a > 1 so that h exchanges the rows in (10.1), that is h(Y i,0 ) = Y i,1 and h(Y i,1 ) = Y i,0 for each i.
Proof. Let Y 1 , . . . , Y q be all the connected components of M \ Z. Denote γ i := Y i ∩ Z. Then by condition (B), γ i is a unique common boundary component of Y i and Z.
Thus {Y i } i=1,...,q splits into pairs which are exchanged by h.
Now it is enough to apply Lemma 9.1 with m = 2. Then there exists g such that g = h on Z and g 2 ∈ S id (f ). Notice that each component of Z contains at least one saddle point of f , otherwise by Lemma 8.2 (4) we have that M = Z. So the first condition (1) of Lemma 9.1 holds. The second condition (2) follows from Lemma 10.1 and the third condition (3) follows from Lemma 10.2. Theorem 3.6 is completed.